Theorem sbcie2s 15916

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

Hypotheses

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

Assertion

Ref	Expression
sbcie2s	|- ( w = W -> ( [. ( E ` w ) / a ]. [. ( F ` w ) / b ]. ps <-> ph ) )

Distinct variable groups:   a, b, w   E, a, b   F, b   W, a, b   ph, a, b

Allowed substitution hints:   ph( w)   ps( w, a, b)   A( w, a, b)   B( w, a, b)   E( w)   F( w, a)   W( w)

Proof of Theorem sbcie2s

Step	Hyp	Ref	Expression
1		fvex 6201	. 2 |- ( E ` w ) e. _V
2		fvex 6201	. 2 |- ( F ` w ) e. _V
3		simprl 794	. . . . . 6 |- ( ( w = W /\ ( a = ( E ` w ) /\ b = ( F ` w ) ) ) -> a = ( E ` w ) )
4		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( E ` w ) = ( E ` W ) )
5		sbcie2s.a	. . . . . . . 8 |- A = ( E ` W )
6	4, 5	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( E ` w ) = A )
7	6	adantr 481	. . . . . 6 |- ( ( w = W /\ ( a = ( E ` w ) /\ b = ( F ` w ) ) ) -> ( E ` w ) = A )
8	3, 7	eqtrd 2656	. . . . 5 |- ( ( w = W /\ ( a = ( E ` w ) /\ b = ( F ` w ) ) ) -> a = A )
9		simprr 796	. . . . . 6 |- ( ( w = W /\ ( a = ( E ` w ) /\ b = ( F ` w ) ) ) -> b = ( F ` w ) )
10		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( F ` w ) = ( F ` W ) )
11		sbcie2s.b	. . . . . . . 8 |- B = ( F ` W )
12	10, 11	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( F ` w ) = B )
13	12	adantr 481	. . . . . 6 |- ( ( w = W /\ ( a = ( E ` w ) /\ b = ( F ` w ) ) ) -> ( F ` w ) = B )
14	9, 13	eqtrd 2656	. . . . 5 |- ( ( w = W /\ ( a = ( E ` w ) /\ b = ( F ` w ) ) ) -> b = B )
15		sbcie2s.1	. . . . 5 |- ( ( a = A /\ b = B ) -> ( ph <-> ps ) )
16	8, 14, 15	syl2anc 693	. . . 4 |- ( ( w = W /\ ( a = ( E ` w ) /\ b = ( F ` w ) ) ) -> ( ph <-> ps ) )
17	16	bicomd 213	. . 3 |- ( ( w = W /\ ( a = ( E ` w ) /\ b = ( F ` w ) ) ) -> ( ps <-> ph ) )
18	17	ex 450	. 2 |- ( w = W -> ( ( a = ( E ` w ) /\ b = ( F ` w ) ) -> ( ps <-> ph ) ) )
19	1, 2, 18	sbc2iedv 3506	1 |- ( w = W -> ( [. ( E ` w ) / a ]. [. ( F ` w ) / b ]. ps <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   [.wsbc 3435   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  istrkgc  25353  istrkgb  25354  istrkge  25356  istrkgl  25357  ishpg  25651  iscgra  25701