%------------------------------------------------------------------------------
% File     : LCL450+2 : TPTP v9.0.0. Released v3.3.0.
% Domain   : Logic Calculi (Propositional)
% Problem  : Congruence of equiv lemmas, to admit substitution of equivalents
% Version  : [HB34] axioms.
% English  :

% Refs     : [HB34]  Hilbert & Bernays (1934), Grundlagen der Mathematick
%          : [Hal]   Halleck (URL), John Halleck's Logic Systems
%          : [Rab06] Rabe (2006), Towards Determining the Subset Relation b
% Source   : [TPTP]
% Names    :

% Status   : Theorem
% Rating   : 0.97 v8.2.0, 1.00 v3.3.0
% Syntax   : Number of formulae    :   49 (  17 unt;   0 def)
%            Number of atoms       :   92 (   6 equ)
%            Maximal formula atoms :   10 (   1 avg)
%            Number of connectives :   43 (   0   ~;   0   |;   6   &)
%                                         (  26 <=>;  11  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   3 avg)
%            Maximal term depth    :    5 (   2 avg)
%            Number of predicates  :   33 (  32 usr;  31 prp; 0-2 aty)
%            Number of functors    :    5 (   5 usr;   0 con; 1-2 aty)
%            Number of variables   :   76 (  76   !;   0   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Difficult and case split into two sub cases
%------------------------------------------------------------------------------
%----Include axioms of propositional logic
include('Axioms/LCL006+0.ax').
include('Axioms/LCL006+1.ax').
%------------------------------------------------------------------------------
%----Include Hilbert's axiomatization of propositional logic, without
%----substitution of equivalents
%----Operator definitions to reduce everything to and & not
fof(hilbert_op_or,axiom,
    op_or ).

fof(hilbert_op_implies_and,axiom,
    op_implies_and ).

fof(hilbert_op_equiv,axiom,
    op_equiv ).

%----The one explicit rule
fof(hilbert_modus_ponens,axiom,
    modus_ponens ).

%----The axioms
fof(hilbert_modus_tollens,axiom,
    modus_tollens ).

fof(hilbert_implies_1,axiom,
    implies_1 ).

fof(hilbert_implies_2,axiom,
    implies_2 ).

fof(hilbert_implies_3,axiom,
    implies_3 ).

fof(hilbert_and_1,axiom,
    and_1 ).

fof(hilbert_and_2,axiom,
    and_2 ).

fof(hilbert_and_3,axiom,
    and_3 ).

fof(hilbert_or_1,axiom,
    or_1 ).

fof(hilbert_or_2,axiom,
    or_2 ).

fof(hilbert_or_3,axiom,
    or_3 ).

fof(hilbert_equivalence_1,axiom,
    equivalence_1 ).

fof(hilbert_equivalence_2,axiom,
    equivalence_2 ).

fof(hilbert_equivalence_3,axiom,
    equivalence_3 ).

fof(equiv_congruence,conjecture,
    ( ! [X] : is_a_theorem(equiv(X,X))
    & ! [X,Y] :
        ( is_a_theorem(equiv(X,Y))
       => is_a_theorem(equiv(not(X),not(Y))) )
    & ! [X1,X2,Y] :
        ( is_a_theorem(equiv(X1,X2))
       => is_a_theorem(equiv(and(X1,Y),and(X2,Y))) )
    & ! [X1,X2,Y] :
        ( is_a_theorem(equiv(X1,X2))
       => is_a_theorem(equiv(and(Y,X1),and(Y,X2))) )
    & ! [X,Y] :
        ( ( is_a_theorem(X)
          & is_a_theorem(equiv(X,Y)) )
       => is_a_theorem(Y) ) ) ).

%------------------------------------------------------------------------------