Theorem sbcies 29322

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

Hypotheses

Ref	Expression
sbcies.a	|- A = ( E ` W )
sbcies.1	|- ( a = A -> ( ph <-> ps ) )

Assertion

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

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

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

Proof of Theorem sbcies

Step	Hyp	Ref	Expression
1		fvexd 6203	. 2 |- ( w = W -> ( E ` w ) e. _V )
2		simpr 477	. . . . 5 |- ( ( w = W /\ a = ( E ` w ) ) -> a = ( E ` w ) )
3		fveq2 6191	. . . . . . 7 |- ( w = W -> ( E ` w ) = ( E ` W ) )
4		sbcies.a	. . . . . . 7 |- A = ( E ` W )
5	3, 4	syl6reqr 2675	. . . . . 6 |- ( w = W -> A = ( E ` w ) )
6	5	adantr 481	. . . . 5 |- ( ( w = W /\ a = ( E ` w ) ) -> A = ( E ` w ) )
7	2, 6	eqtr4d 2659	. . . 4 |- ( ( w = W /\ a = ( E ` w ) ) -> a = A )
8		sbcies.1	. . . 4 |- ( a = A -> ( ph <-> ps ) )
9	7, 8	syl 17	. . 3 |- ( ( w = W /\ a = ( E ` w ) ) -> ( ph <-> ps ) )
10	9	bicomd 213	. 2 |- ( ( w = W /\ a = ( E ` w ) ) -> ( ps <-> ph ) )
11	1, 10	sbcied 3472	1 |- ( w = W -> ( [. ( E ` w ) / a ]. ps <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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: (None)