Theorem cbvraldva2 2749

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 110	. . . . 5 |- ( ( ph /\ x = y ) -> x = y )
2		cbvraldva2.2	. . . . 5 |- ( ( ph /\ x = y ) -> A = B )
3	1, 2	eleq12d 2278	. . . 4 |- ( ( ph /\ x = y ) -> ( x e. A <-> y e. B ) )
4		cbvraldva2.1	. . . 4 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
5	3, 4	imbi12d 234	. . 3 |- ( ( ph /\ x = y ) -> ( ( x e. A -> ps ) <-> ( y e. B -> ch ) ) )
6	5	cbvaldva 1953	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) <-> A. y ( y e. B -> ch ) ) )
7		df-ral 2491	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
8		df-ral 2491	. 2 |- ( A. y e. B ch <-> A. y ( y e. B -> ch ) )
9	6, 7, 8	3bitr4g 223	1 |- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2178   A.wral 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-cleq 2200  df-clel 2203  df-ral 2491

This theorem is referenced by:  cbvraldva  2751  acexmid  5966  tfrlem3ag  6418  tfrlem3a  6419  tfrlemi1  6441  tfr1onlem3ag  6446