Theorem sbcie3s 15917

Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)

Hypotheses

Ref	Expression
sbcie3s.a	|- A = ( E ` W )
sbcie3s.b	|- B = ( F ` W )
sbcie3s.c	|- C = ( G ` W )
sbcie3s.1	|- ( ( a = A /\ b = B /\ c = C ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbcie3s	|- ( w = W -> ( [. ( E ` w ) / a ]. [. ( F ` w ) / b ]. [. ( G ` w ) / c ]. ps <-> ph ) )

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

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

Proof of Theorem sbcie3s

Step	Hyp	Ref	Expression
1		fvexd 6203	. 2 |- ( w = W -> ( E ` w ) e. _V )
2		fvexd 6203	. . 3 |- ( ( w = W /\ a = ( E ` w ) ) -> ( F ` w ) e. _V )
3		fvexd 6203	. . . 4 |- ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) -> ( G ` w ) e. _V )
4		simpllr 799	. . . . . . . 8 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> a = ( E ` w ) )
5		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( E ` w ) = ( E ` W ) )
6	5	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> ( E ` w ) = ( E ` W ) )
7	4, 6	eqtrd 2656	. . . . . . 7 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> a = ( E ` W ) )
8		sbcie3s.a	. . . . . . 7 |- A = ( E ` W )
9	7, 8	syl6eqr 2674	. . . . . 6 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> a = A )
10		simplr 792	. . . . . . . 8 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> b = ( F ` w ) )
11		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( F ` w ) = ( F ` W ) )
12	11	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> ( F ` w ) = ( F ` W ) )
13	10, 12	eqtrd 2656	. . . . . . 7 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> b = ( F ` W ) )
14		sbcie3s.b	. . . . . . 7 |- B = ( F ` W )
15	13, 14	syl6eqr 2674	. . . . . 6 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> b = B )
16		simpr 477	. . . . . . . 8 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> c = ( G ` w ) )
17		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( G ` w ) = ( G ` W ) )
18	17	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> ( G ` w ) = ( G ` W ) )
19	16, 18	eqtrd 2656	. . . . . . 7 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> c = ( G ` W ) )
20		sbcie3s.c	. . . . . . 7 |- C = ( G ` W )
21	19, 20	syl6eqr 2674	. . . . . 6 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> c = C )
22		sbcie3s.1	. . . . . 6 |- ( ( a = A /\ b = B /\ c = C ) -> ( ph <-> ps ) )
23	9, 15, 21, 22	syl3anc 1326	. . . . 5 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> ( ph <-> ps ) )
24	23	bicomd 213	. . . 4 |- ( ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) /\ c = ( G ` w ) ) -> ( ps <-> ph ) )
25	3, 24	sbcied 3472	. . 3 |- ( ( ( w = W /\ a = ( E ` w ) ) /\ b = ( F ` w ) ) -> ( [. ( G ` w ) / c ]. ps <-> ph ) )
26	2, 25	sbcied 3472	. 2 |- ( ( w = W /\ a = ( E ` w ) ) -> ( [. ( F ` w ) / b ]. [. ( G ` w ) / c ]. ps <-> ph ) )
27	1, 26	sbcied 3472	1 |- ( w = W -> ( [. ( E ` w ) / a ]. [. ( F ` w ) / b ]. [. ( G ` w ) / c ]. ps <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   _Vcvv 3200   [.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:  istrkgcb  25355  istrkgld  25358  legval  25479  istrkg2d  30744  afsval  30749