Theorem rsp2 2600

Description: Restricted specialization. (Contributed by NM, 11-Feb-1997.)

Assertion

Ref	Expression
rsp2	|- ( A. x e. A A. y e. B ph -> ( ( x e. A /\ y e. B ) -> ph ) )

Proof of Theorem rsp2

Step	Hyp	Ref	Expression
1		rsp 2597	. . 3 |- ( A. x e. A A. y e. B ph -> ( x e. A -> A. y e. B ph ) )
2		rsp 2597	. . 3 |- ( A. y e. B ph -> ( y e. B -> ph ) )
3	1, 2	syl6 33	. 2 |- ( A. x e. A A. y e. B ph -> ( x e. A -> ( y e. B -> ph ) ) )
4	3	impd 254	1 |- ( A. x e. A A. y e. B ph -> ( ( x e. A /\ y e. B ) -> ph ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   A.wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-4 1563

This proof depends on definitions:  df-bi 117  df-ral 2533

This theorem is used by:  ralidm  3628  sowlin  4465  cmncom  14105  cnmpt21  15392  cnmpt2t  15394  cnmpt22  15395  cnmptcom  15399