Theorem 2ralbidva 1678

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

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

Proof of Theorem 2ralbidva

Step	Hyp	Ref	Expression
1		ax-17 971	. 2 |- (ph -> A.xph)
2		ax-17 971	. 2 |- (ph -> A.yph)
3		2ralbidva.1	. 2 |- ((ph /\ (x e. A /\ y e. B)) -> (ps <-> ch))
4	1, 2, 3	2ralbida 1677	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   e. wcel 958  A.wral 1645

This theorem is referenced by:  isowe 3903  f1oweALT 3906  ghomgsg 10395

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1649

Copyright terms: Public domain