Theorem bi2ralda 1232

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

Hypotheses

Ref	Expression
bi2ralda.1	|- (ph -> A.xph)
bi2ralda.2	|- (ph -> A.yph)
bi2ralda.3	|- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))

Assertion

Ref	Expression
bi2ralda	|- (ph -> (A.x e. A A.y e. B ps <-> A.x e. A A.y e. B ch))

Distinct variable group(s):   x,y   y,A

Proof of Theorem bi2ralda

Step	Hyp	Ref	Expression
1		bi2ralda.1	. 2 |- (ph -> A.xph)
2		bi2ralda.2	. . . 4 |- (ph -> A.yph)
3		ax-17 925	. . . 4 |- (x e. A -> A.y x e. A)
4	2, 3	hban 704	. . 3 |- ((ph /\ x e. A) -> A.y(ph /\ x e. A))
5		bi2ralda.3	. . . 4 |- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))
6	5	anassrs 338	. . 3 |- (((ph /\ x e. A) /\ y e. B) -> (ps <-> ch))
7	4, 6	biralda 1213	. 2 |- ((ph /\ x e. A) -> (A.y e. B ps <-> A.y e. B ch))
8	1, 7	biralda 1213	1 |- (ph -> (A.x e. A A.y e. B ps <-> A.x e. A A.y e. B ch))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  bi2raldva 1233

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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