Theorem 2rexbidv 2658

Description: Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)

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

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

Proof of Theorem 2rexbidv

Step	Hyp	Ref	Expression
1		2ralbidv.1	. . 3 |- (ph -> (ps <-> ch))
2	1	rexbidv 2636	. 2 |- (ph -> (E.y e. B ps <-> E.y e. B ch))
3	2	rexbidv 2636	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  E.wrex 2616

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 2621

This theorem is referenced by:  eladdc  4399  f1oiso  5500  ovelrn  5609  mucex  6134