Theorem rspec2 2559

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

Hypothesis

Ref	Expression
rspec2.1	|- A. x e. A A. y e. B ph

Assertion

Ref	Expression
rspec2	|- ( ( x e. A /\ y e. B ) -> ph )

Proof of Theorem rspec2

Step	Hyp	Ref	Expression
1		rspec2.1	. . 3 |- A. x e. A A. y e. B ph
2	1	rspec 2522	. 2 |- ( x e. A -> A. y e. B ph )
3	2	r19.21bi 2558	1 |- ( ( x e. A /\ y e. B ) -> ph )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-ral 2453

This theorem is referenced by:  rspec3  2560  ordtriexmid  4505  onsucsssucexmid  4511