Theorem 2ralbidva 2572

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 1581	. 2 |- F/ x ph
2		nfv 1581	. 2 |- F/ y ph
3		2ralbidva.1	. 2 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) )
4	1, 2, 3	2ralbida 2571	1 |- ( ph -> ( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   A.wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533

This theorem is used by:  soinxp  4845  isotr  6022  fnmpoovd  6451  sgrppropd  13777  mndpropd  13802  mhmpropd  13822  cmnpropd  14147  rngpropd  14303  ringpropd  14392  lmodprop2d  14734  lsspropdg  14817  assapropd  15063  ismet2  15504  txmetcn  15669