Theorem rexbii2 1672

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 1051	. 2 |- (E.x(x e. A /\ ph) <-> E.x(x e. B /\ ps))
3		df-rex 1650	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4		df-rex 1650	. 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 958  E.wex 980  E.wrex 1646

This theorem is referenced by:  rexbiia 1674  wefrc 2943  bnd2 4724  infm3 6054  infmsup 6068  rexuz2 6445  infpn2 7509  blrn2 7842  sumdmdi 10342

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-rex 1650

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