Theorem 2ralbidva 2552

Description: Formula-building rule for restricted universal quantifiers (deduction form). (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 1574	. 2 |- F/ x ph
2		nfv 1574	. 2 |- F/ y ph
3		2ralbidva.1	. 2 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) )
4	1, 2, 3	2ralbida 2551	1 |- ( ph -> ( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   A.wral 2508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-4 1556  ax-17 1572

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513

This theorem is referenced by:  soinxp  4789  isotr  5940  fnmpoovd  6361  sgrppropd  13446  mndpropd  13473  mhmpropd  13499  cmnpropd  13832  rngpropd  13918  ringpropd  14001  lmodprop2d  14312  lsspropdg  14395  ismet2  15028  txmetcn  15193