Theorem cbvrexdva2 2737

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 110	. . . . 5 |- ( ( ph /\ x = y ) -> x = y )
2		cbvraldva2.2	. . . . 5 |- ( ( ph /\ x = y ) -> A = B )
3	1, 2	eleq12d 2267	. . . 4 |- ( ( ph /\ x = y ) -> ( x e. A <-> y e. B ) )
4		cbvraldva2.1	. . . 4 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
5	3, 4	anbi12d 473	. . 3 |- ( ( ph /\ x = y ) -> ( ( x e. A /\ ps ) <-> ( y e. B /\ ch ) ) )
6	5	cbvexdva 1944	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. y ( y e. B /\ ch ) ) )
7		df-rex 2481	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
8		df-rex 2481	. 2 |- ( E. y e. B ch <-> E. y ( y e. B /\ ch ) )
9	6, 7, 8	3bitr4g 223	1 |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-cleq 2189  df-clel 2192  df-rex 2481

This theorem is referenced by:  cbvrexdva  2739  acexmid  5921