Theorem rspec2 2934

Description: Specialization rule for restricted quantification, with two quantifiers. (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 2931	. 2 |- ( x e. A -> A. y e. B ph )
3	2	r19.21bi 2932	1 |- ( ( x e. A /\ y e. B ) -> ph )

Syntax hints:   -> wi 4   /\ wa 384   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-ex 1705  df-ral 2917

This theorem is referenced by:  rspec3  2935