Theorem sbcie2s

Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019) (Revised by SN, 2-Mar-2025)

	Ref	Expression
Hypotheses	sbcie2s.a	|- A = ( E ` W )
	sbcie2s.b	|- B = ( F ` W )
	sbcie2s.1	|- ( ( a = A /\ b = B ) -> ( ph <-> ps ) )
Assertion	sbcie2s	|- ( w = W -> ( [. ( E ` w ) / a ]. [. ( F ` w ) / b ]. ph <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		sbcie2s.a	|- A = ( E ` W )
2		sbcie2s.b	|- B = ( F ` W )
3		sbcie2s.1	|- ( ( a = A /\ b = B ) -> ( ph <-> ps ) )
4		fvex	|- ( E ` w ) e. _V
5		fvex	|- ( F ` w ) e. _V
6		fveq2	|- ( w = W -> ( E ` w ) = ( E ` W ) )
7	6 1	eqtr4di	|- ( w = W -> ( E ` w ) = A )
8	7	eqeq2d	|- ( w = W -> ( a = ( E ` w ) <-> a = A ) )
9	8	biimpd	|- ( w = W -> ( a = ( E ` w ) -> a = A ) )
10		fveq2	|- ( w = W -> ( F ` w ) = ( F ` W ) )
11	10 2	eqtr4di	|- ( w = W -> ( F ` w ) = B )
12	11	eqeq2d	|- ( w = W -> ( b = ( F ` w ) <-> b = B ) )
13	12	biimpd	|- ( w = W -> ( b = ( F ` w ) -> b = B ) )
14	3	a1i	|- ( w = W -> ( ( a = A /\ b = B ) -> ( ph <-> ps ) ) )
15	9 13 14	syl2and	|- ( w = W -> ( ( a = ( E ` w ) /\ b = ( F ` w ) ) -> ( ph <-> ps ) ) )
16	4 5 15	sbc2iedv	|- ( w = W -> ( [. ( E ` w ) / a ]. [. ( F ` w ) / b ]. ph <-> ps ) )