Theorem r19.21advv 2184

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

Hypothesis

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

Assertion

Ref	Expression
r19.21advv	|- (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 r19.21advv

Step	Hyp	Ref	Expression
1		r19.21advv.1	. . . 4 |- (ph -> (ps -> ((x e. A /\ y e. B) -> ch)))
2	1	imp 377	. . 3 |- ((ph /\ ps) -> ((x e. A /\ y e. B) -> ch))
3	2	r19.21aivv 2183	. 2 |- ((ph /\ ps) -> A.x e. A A.y e. B ch)
4	3	ex 402	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:  r19.21advva 2185  tgss2 8907  hausfillim 10303  connsub 15443  ist1-2 15542  ist1-3 15543  clatlat 16893

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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