Theorem 2rexbidva 2655

Description: Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 15-Dec-2004.)

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

Allowed substitution hints:   ps(x,y)   ch(x,y)   A(x)   B(x,y)

Proof of Theorem 2rexbidva

Step	Hyp	Ref	Expression
1		2ralbidva.1	. . . 4 |- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))
2	1	anassrs 629	. . 3 |- (((ph /\ x e. A) /\ y e. B) -> (ps <-> ch))
3	2	rexbidva 2631	. 2 |- ((ph /\ x e. A) -> (E.y e. B ps <-> E.y e. B ch))
4	3	rexbidva 2631	1 |- (ph -> (E.x e. A E.y e. B ps <-> E.x e. A E.y e. B ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   e. wcel 1710  E.wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-rex 2620

This theorem is referenced by: (None)