Theorem 2ralbida 2491

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 24-Feb-2004.)

Hypotheses

Ref	Expression
2ralbida.1	|- F/ x ph
2ralbida.2	|- F/ y ph
2ralbida.3	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) )

Assertion

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

Distinct variable groups:   x, y   y, A

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

Proof of Theorem 2ralbida

Step	Hyp	Ref	Expression
1		2ralbida.1	. 2 |- F/ x ph
2		2ralbida.2	. . . 4 |- F/ y ph
3		nfv 1521	. . . 4 |- F/ y x e. A
4	2, 3	nfan 1558	. . 3 |- F/ y ( ph /\ x e. A )
5		2ralbida.3	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) )
6	5	anassrs 398	. . 3 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps <-> ch ) )
7	4, 6	ralbida 2464	. 2 |- ( ( ph /\ x e. A ) -> ( A. y e. B ps <-> A. y e. B ch ) )
8	1, 7	ralbida 2464	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 103   <-> wb 104   F/wnf 1453   e. wcel 2141   A.wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453

This theorem is referenced by:  2ralbidva  2492