let x be bound_QC-variable; :: thesis: for p being Element of QC-WFF st p is conjunctive holds
p . x = ((the_left_argument_of p) . x) '&' ((the_right_argument_of p) . x)
let p be Element of QC-WFF ; :: thesis: ( p is conjunctive implies p . x = ((the_left_argument_of p) . x) '&' ((the_right_argument_of p) . x) )
consider F being Function of QC-WFF ,QC-WFF such that
A1: p . x = F . p and
A2: for q being Element of QC-WFF holds
( F . VERUM = VERUM & ( q is atomic implies F . q = (the_pred_symbol_of q) ! (Subst (the_arguments_of q),((a. 0 ) .--> x)) ) & ( q is negative implies F . q = 'not' (F . (the_argument_of q)) ) & ( q is conjunctive implies F . q = (F . (the_left_argument_of q)) '&' (F . (the_right_argument_of q)) ) & ( q is universal implies F . q = IFEQ (bound_in q),x,q,(All (bound_in q),(F . (the_scope_of q))) ) ) by Def6;
consider F1 being Function of QC-WFF ,QC-WFF such that
A3: (the_left_argument_of p) . x = F1 . (the_left_argument_of p) and
A4: for q being Element of QC-WFF holds
( F1 . VERUM = VERUM & ( q is atomic implies F1 . q = (the_pred_symbol_of q) ! (Subst (the_arguments_of q),((a. 0 ) .--> x)) ) & ( q is negative implies F1 . q = 'not' (F1 . (the_argument_of q)) ) & ( q is conjunctive implies F1 . q = (F1 . (the_left_argument_of q)) '&' (F1 . (the_right_argument_of q)) ) & ( q is universal implies F1 . q = IFEQ (bound_in q),x,q,(All (bound_in q),(F1 . (the_scope_of q))) ) ) by Def6;
consider F2 being Function of QC-WFF ,QC-WFF such that
A5: (the_right_argument_of p) . x = F2 . (the_right_argument_of p) and
A6: for q being Element of QC-WFF holds
( F2 . VERUM = VERUM & ( q is atomic implies F2 . q = (the_pred_symbol_of q) ! (Subst (the_arguments_of q),((a. 0 ) .--> x)) ) & ( q is negative implies F2 . q = 'not' (F2 . (the_argument_of q)) ) & ( q is conjunctive implies F2 . q = (F2 . (the_left_argument_of q)) '&' (F2 . (the_right_argument_of q)) ) & ( q is universal implies F2 . q = IFEQ (bound_in q),x,q,(All (bound_in q),(F2 . (the_scope_of q))) ) ) by Def6;
( F1 = F & F2 = F ) by A2, A4, A6, Lm2;
hence ( p is conjunctive implies p . x = ((the_left_argument_of p) . x) '&' ((the_right_argument_of p) . x) ) by A1, A2, A3, A5; :: thesis: verum