Theorem cbvraldva2 3175

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 477	. . . . 5 |- ( ( ph /\ x = y ) -> x = y )
2		cbvraldva2.2	. . . . 5 |- ( ( ph /\ x = y ) -> A = B )
3	1, 2	eleq12d 2695	. . . 4 |- ( ( ph /\ x = y ) -> ( x e. A <-> y e. B ) )
4		cbvraldva2.1	. . . 4 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
5	3, 4	imbi12d 334	. . 3 |- ( ( ph /\ x = y ) -> ( ( x e. A -> ps ) <-> ( y e. B -> ch ) ) )
6	5	cbvaldva 2281	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) <-> A. y ( y e. B -> ch ) ) )
7		df-ral 2917	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
8		df-ral 2917	. 2 |- ( A. y e. B ch <-> A. y ( y e. B -> ch ) )
9	6, 7, 8	3bitr4g 303	1 |- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912

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-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ral 2917

This theorem is referenced by:  cbvraldva  3177  tfrlem3a  7473  mreexexlemd  16304