Theorem cbvraldva2 2581

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
cbvraldva2.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
cbvraldva2.2	|- ( ( ph /\ x = y ) -> A = B )

Assertion

Ref	Expression
cbvraldva2	|- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) )

Distinct variable groups:   y, A   ps, y   x, B   ch, x   ph, x, y

Allowed substitution hints:   ps( x)   ch( y)   A( x)   B( y)

Proof of Theorem cbvraldva2

Step	Hyp	Ref	Expression
1		simpr 108	. . . . 5 |- ( ( ph /\ x = y ) -> x = y )
2		cbvraldva2.2	. . . . 5 |- ( ( ph /\ x = y ) -> A = B )
3	1, 2	eleq12d 2149	. . . 4 |- ( ( ph /\ x = y ) -> ( x e. A <-> y e. B ) )
4		cbvraldva2.1	. . . 4 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
5	3, 4	imbi12d 232	. . 3 |- ( ( ph /\ x = y ) -> ( ( x e. A -> ps ) <-> ( y e. B -> ch ) ) )
6	5	cbvaldva 1844	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) <-> A. y ( y e. B -> ch ) ) )
7		df-ral 2353	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
8		df-ral 2353	. 2 |- ( A. y e. B ch <-> A. y ( y e. B -> ch ) )
9	6, 7, 8	3bitr4g 221	1 |- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077  df-ral 2353

This theorem is referenced by:  cbvraldva  2583  acexmid  5531  tfrlem3ag  5947  tfrlem3a  5948  tfrlemi1  5969