Theorem alral 2452

Description: Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)

Assertion

Ref	Expression
alral	|- ( A. x ph -> A. x e. A ph )

Proof of Theorem alral

Step	Hyp	Ref	Expression
1		ax-1 5	. . 3 |- ( ph -> ( x e. A -> ph ) )
2	1	alimi 1414	. 2 |- ( A. x ph -> A. x ( x e. A -> ph ) )
3		df-ral 2395	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4	2, 3	sylibr 133	1 |- ( A. x ph -> A. x e. A ph )

Syntax hints:   -> wi 4   A.wal 1312   e. wcel 1463   A.wral 2390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408

This theorem depends on definitions:  df-bi 116  df-ral 2395

This theorem is referenced by:  abnex  4328  find  4473  findset  12835