Theorem rspec3 2556

Description: Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)

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 2555	. . 3 |- ( ( x e. A /\ y e. B ) -> A. z e. C ph )
3	2	r19.21bi 2554	. 2 |- ( ( ( x e. A /\ y e. B ) /\ z e. C ) -> ph )
4	3	3impa 1184	1 |- ( ( x e. A /\ y e. B /\ z e. C ) -> ph )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   e. wcel 2136   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-3an 970  df-ral 2449

This theorem is referenced by:  ordsoexmid  4539