Theorem 2rexbidva 1679

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

Hypothesis

Ref	Expression
2ralbidva.1	|- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))

Assertion

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

Distinct variable groups:   x,y,ph   y,A

Proof of Theorem 2rexbidva

Step	Hyp	Ref	Expression
1		2ralbidva.1	. . . 4 |- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))
2	1	anassrs 441	. . 3 |- (((ph /\ x e. A) /\ y e. B) -> (ps <-> ch))
3	2	rexbidva 1660	. 2 |- ((ph /\ x e. A) -> (E.y e. B ps <-> E.y e. B ch))
4	3	rexbidva 1660	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 146   /\ wa 223   e. wcel 958  E.wrex 1646

This theorem is referenced by:  shscomt 9283

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  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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