Theorem ra42 1696

Description: Restricted specialization.

Assertion

Ref	Expression
ra42	|- (A.x e. A A.y e. B ph -> ((x e. A /\ y e. B) -> ph))

Proof of Theorem ra42

Step	Hyp	Ref	Expression
1		ra4 1694	. . 3 |- (A.x e. A A.y e. B ph -> (x e. A -> A.y e. B ph))
2		ra4 1694	. . 3 |- (A.y e. B ph -> (y e. B -> ph))
3	1, 2	syl6 22	. 2 |- (A.x e. A A.y e. B ph -> (x e. A -> (y e. B -> ph)))
4	3	imp3a 361	1 |- (A.x e. A A.y e. B ph -> ((x e. A /\ y e. B) -> ph))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  A.wral 1645

This theorem is referenced by:  solin 2857  ralxp 3218  f1fveq 3876  isotrALT 3898  elrnoprabg 4124  subgabl 8123  ismonc 10742

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 973

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1649

Copyright terms: Public domain