Theorem cbvrexdva2 3176

Description: Rule used to change the bound variable in a restricted existential 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
cbvrexdva2	|- ( ph -> ( E. x e. A ps <-> E. 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 cbvrexdva2

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	anbi12d 747	. . 3 |- ( ( ph /\ x = y ) -> ( ( x e. A /\ ps ) <-> ( y e. B /\ ch ) ) )
6	5	cbvexdva 2283	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. y ( y e. B /\ ch ) ) )
7		df-rex 2918	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
8		df-rex 2918	. 2 |- ( E. y e. B ch <-> E. y ( y e. B /\ ch ) )
9	6, 7, 8	3bitr4g 303	1 |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913

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-rex 2918

This theorem is referenced by:  cbvrexdva  3178  mreexexlemd  16304  eulerpartlemgvv  30438