Theorem 2ralbida 1669

Description: Formula-building rule for restricted universal quantifier (deduction rule).

Hypotheses

Ref	Expression
2ralbida.1	|- (ph -> A.xph)
2ralbida.2	|- (ph -> A.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

Proof of Theorem 2ralbida

Step	Hyp	Ref	Expression
1		2ralbida.1	. 2 |- (ph -> A.xph)
2		2ralbida.2	. . . 4 |- (ph -> A.yph)
3		ax-17 968	. . . 4 |- (x e. A -> A.y x e. A)
4	2, 3	hban 1006	. . 3 |- ((ph /\ x e. A) -> A.y(ph /\ x e. A))
5		2ralbida.3	. . . 4 |- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))
6	5	anassrs 441	. . 3 |- (((ph /\ x e. A) /\ y e. B) -> (ps <-> ch))
7	4, 6	ralbida 1649	. 2 |- ((ph /\ x e. A) -> (A.y e. B ps <-> A.y e. B ch))
8	1, 7	ralbida 1649	1 |- (ph -> (A.x e. A A.y e. B ps <-> A.x e. A A.y e. B ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   e. wcel 955  A.wral 1637

This theorem is referenced by:  2ralbidva 1670

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1641

Copyright terms: Public domain