Theorem bi2rexdva 1234

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

Hypothesis

Ref	Expression
bi2raldva.1	|- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))

Assertion

Ref	Expression
bi2rexdva	|- (ph -> (E.x e. A E.y e. B ps <-> E.x e. A E.y e. B ch))

Distinct variable group(s):   x,y,ph   y,A

Proof of Theorem bi2rexdva

Step	Hyp	Ref	Expression
1		bi2raldva.1	. . . 4 |- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))
2	1	anassrs 338	. . 3 |- (((ph /\ x e. A) /\ y e. B) -> (ps <-> ch))
3	2	birexdva 1216	. 2 |- ((ph /\ x e. A) -> (E.y e. B ps <-> E.y e. B ch))
4	3	birexdva 1216	1 |- (ph -> (E.x e. A E.y e. B ps <-> E.x e. A E.y e. B ch))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  shscomt 5284

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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