Theorem rspec3 1268

Description: Specialization rule for restricted quantification.

Hypothesis

Ref	Expression
rspec3.1	|- A.x e. A A.y e. B A.z e. C ph

Assertion

Ref	Expression
rspec3	|- ((x e. A /\ y e. B /\ z e. C) -> ph)

Proof of Theorem rspec3

Step	Hyp	Ref	Expression
1		rspec3.1	. . . 4 |- A.x e. A A.y e. B A.z e. C ph
2	1	rspec2 1267	. . 3 |- ((x e. A /\ y e. B) -> A.z e. C ph)
3	2	r19.21bi 1266	. 2 |- (((x e. A /\ y e. B) /\ z e. C) -> ph)
4	3	3impa 609	1 |- ((x e. A /\ y e. B /\ z e. C) -> ph)

Syntax hints:   -> wi 2   /\ wa 196   /\ w3a 581   e. wcel 1092  A.wral 1201

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ral 1205

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