Theorem r19.21aivva 15653

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.)

Hypothesis

Ref	Expression
r19.21aivva.1	|- ((ph /\ (x e. A /\ y e. B)) -> ps)

Assertion

Ref	Expression
r19.21aivva	|- (ph -> A.x e. A A.y e. B ps)

Distinct variable groups:   ph,x,y   y,A

Proof of Theorem r19.21aivva

Step	Hyp	Ref	Expression
1		r19.21aivva.1	. . 3 |- ((ph /\ (x e. A /\ y e. B)) -> ps)
2	1	ex 402	. 2 |- (ph -> ((x e. A /\ y e. B) -> ps))
3	2	r19.21aivv 2183	1 |- (ph -> A.x e. A A.y e. B ps)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  A.wral 2105

This theorem is referenced by:  grpkerinj 16042  isdivrng2 16111  rnggrphom 16125  rnghomco 16128  rngisocnv 16135  linepsub 17232  pmapsub 17250

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-an 242  df-ral 2109

Copyright terms: Public domain