Theorem r19.21advva 2185

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

Hypothesis

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

Assertion

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

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

Proof of Theorem r19.21advva

Step	Hyp	Ref	Expression
1		r19.21advva.1	. . . 4 |- ((ph /\ (x e. A /\ y e. B)) -> (ps -> ch))
2	1	ex 402	. . 3 |- (ph -> ((x e. A /\ y e. B) -> (ps -> ch)))
3	2	com23 36	. 2 |- (ph -> (ps -> ((x e. A /\ y e. B) -> ch)))
4	3	r19.21advv 2184	1 |- (ph -> (ps -> A.x e. A A.y e. B ch))

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

This theorem is referenced by:  basgen2 8909  fbunfip 10282  cdj3i 12013  reconnlem1 15446  reconn 15451  ispridl2 16186  ispridlc 16218

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

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