Theorem rspec3 1727

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 1726	. . 3 |- ((x e. A /\ y e. B) -> A.z e. C ph)
3	2	r19.21bi 1725	. 2 |- (((x e. A /\ y e. B) /\ z e. C) -> ph)
4	3	3impa 828	1 |- ((x e. A /\ y e. B /\ z e. C) -> ph)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   e. wcel 958  A.wral 1645

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 973

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ral 1649

Copyright terms: Public domain