Theorem 2ralbidva 2654

Description: Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)

Hypothesis

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

Assertion

Ref	Expression
2ralbidva	|- (ph -> (A.x e. A A.y e. B ps <-> A.x e. A A.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 2ralbidva

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 |- F/xph
2		nfv 1619	. 2 |- F/yph
3		2ralbidva.1	. 2 |- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))
4	1, 2, 3	2ralbida 2653	1 |- (ph -> (A.x e. A A.y e. B ps <-> A.x e. A A.y e. B ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   e. wcel 1710  A.wral 2614

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-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2619

This theorem is referenced by: (None)