Theorem rexbii2 1669

Description: Inference adding different restricted existential quantifiers to each side of an equivalence.

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 1049	. 2 |- (E.x(x e. A /\ ph) <-> E.x(x e. B /\ ps))
3		df-rex 1647	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4		df-rex 1647	. 2 |- (E.x e. B ps <-> E.x(x e. B /\ ps))
5	2, 3, 4	3bitr4 183	1 |- (E.x e. A ph <-> E.x e. B ps)

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 956  E.wex 978  E.wrex 1643

This theorem is referenced by:  rexbiia 1671  wefrc 2938  bnd2 4704  infm3 6009  infmsup 6023  rexuz2 6385  infpn2 7460  blrn2 7794  sumdmdi 10278

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-rex 1647

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