Theorem r19.21aivv 1263

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

Hypothesis

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

Assertion

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

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

Proof of Theorem r19.21aivv

Step	Hyp	Ref	Expression
1		r19.21aivv.1	. . . 4 |- (ph -> ((x e. A /\ y e. B) -> ps))
2	1	exp3a 292	. . 3 |- (ph -> (x e. A -> (y e. B -> ps)))
3	2	r19.21adv 1262	. 2 |- (ph -> (x e. A -> A.y e. B ps))
4	3	r19.21aiv 1259	1 |- (ph -> A.x e. A A.y e. B ps)

Syntax hints:   -> wi 2   /\ wa 196   e. wcel 1092  A.wral 1201

This theorem is referenced by:  dom2d 3307  uzwo2 4606

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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