Theorem rexbii2 2644

Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)

Hypothesis

Ref	Expression
rexbii2.1	|- ((x e. A /\ ph) <-> (x e. B /\ ps))

Assertion

Ref	Expression
rexbii2	|- (E.x e. A ph <-> E.x e. B ps)

Proof of Theorem rexbii2

Step	Hyp	Ref	Expression
1		rexbii2.1	. . 3 |- ((x e. A /\ ph) <-> (x e. B /\ ps))
2	1	exbii 1582	. 2 |- (E.x(x e. A /\ ph) <-> E.x(x e. B /\ ps))
3		df-rex 2621	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4		df-rex 2621	. 2 |- (E.x e. B ps <-> E.x(x e. B /\ ps))
5	2, 3, 4	3bitr4i 268	1 |- (E.x e. A ph <-> E.x e. B ps)

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   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

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-rex 2621

This theorem is referenced by:  rexeqbii  2646  rexbiia  2648  rexrab  3001  rexdifsn  3844  pw1in  4165