Theorem cbvrexdva2 2841

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 447	. . . . 5 |- ((ph /\ x = y) -> x = y)
2		cbvraldva2.2	. . . . 5 |- ((ph /\ x = y) -> A = B)
3	1, 2	eleq12d 2421	. . . 4 |- ((ph /\ x = y) -> (x e. A <-> y e. B))
4		cbvraldva2.1	. . . 4 |- ((ph /\ x = y) -> (ps <-> ch))
5	3, 4	anbi12d 691	. . 3 |- ((ph /\ x = y) -> ((x e. A /\ ps) <-> (y e. B /\ ch)))
6	5	cbvexdva 2011	. 2 |- (ph -> (E.x(x e. A /\ ps) <-> E.y(y e. B /\ ch)))
7		df-rex 2621	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
8		df-rex 2621	. 2 |- (E.y e. B ch <-> E.y(y e. B /\ ch))
9	6, 7, 8	3bitr4g 279	1 |- (ph -> (E.x e. A ps <-> E.y e. B ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-rex 2621

This theorem is referenced by:  cbvrexdva  2843