Theorem rspec3 2935

Description: Specialization rule for restricted quantification, with three quantifiers. (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 2934	. . 3 |- ( ( x e. A /\ y e. B ) -> A. z e. C ph )
3	2	r19.21bi 2932	. 2 |- ( ( ( x e. A /\ y e. B ) /\ z e. C ) -> ph )
4	3	3impa 1259	1 |- ( ( x e. A /\ y e. B /\ z e. C ) -> ph )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705  df-ral 2917

This theorem is referenced by: (None)