Theorem alral 2515

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 6	. . 3 |- ( ph -> ( x e. A -> ph ) )
2	1	alimi 1448	. 2 |- ( A. x ph -> A. x ( x e. A -> ph ) )
3		df-ral 2453	. 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 1346   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-ia3 107  ax-5 1440  ax-gen 1442

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

This theorem is referenced by:  abnex  4432  find  4583  prodeq2w  11519  findset  13980