Theorem 2ralbida 2654

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

Hypotheses

Ref	Expression
2ralbida.1	|- F/xph
2ralbida.2	|- F/yph
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/xph
2		2ralbida.2	. . . 4 |- F/yph
3		nfv 1619	. . . 4 |- F/y x e. A
4	2, 3	nfan 1824	. . 3 |- F/y(ph /\ x e. A)
5		2ralbida.3	. . . 4 |- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))
6	5	anassrs 629	. . 3 |- (((ph /\ x e. A) /\ y e. B) -> (ps <-> ch))
7	4, 6	ralbida 2629	. 2 |- ((ph /\ x e. A) -> (A.y e. B ps <-> A.y e. B ch))
8	1, 7	ralbida 2629	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   F/wnf 1544   e. wcel 1710  A.wral 2615

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 2620

This theorem is referenced by:  2ralbidva  2655