Theorem 2rexbidv 1728

Description: Formula-building rule for restricted existential quantifiers (deduction rule).

Hypothesis

Ref	Expression
2ralbidv.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
2rexbidv	|- (ph -> (E.x e. A E.y e. B ps <-> E.x e. A E.y e. B ch))

Distinct variable groups:   ph,x   ph,y

Proof of Theorem 2rexbidv

Step	Hyp	Ref	Expression
1		2ralbidv.1	. . 3 |- (ph -> (ps <-> ch))
2	1	rexbidv 1711	. 2 |- (ph -> (E.y e. B ps <-> E.y e. B ch))
3	2	rexbidv 1711	1 |- (ph -> (E.x e. A E.y e. B ps <-> E.x e. A E.y e. B ch))

Syntax hints:   -> wi 3   <-> wb 153  E.wrex 1693

This theorem is referenced by:  f1oiso 3962  oprvalelrn 4097  brdom7disj 4866  brdom6disj 4867  axcnre 5351  elq 6309  hausnei 7869  pjth 9317  shsel 9361  iseuctopg 10596

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-17 1012  ax-4 1014  ax-5o 1016

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-rex 1697

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