Theorem alral 2928

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		ala1 1741	. 2 |- ( A. x ph -> A. x ( x e. A -> ph ) )
2		df-ral 2917	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3	1, 2	sylibr 224	1 |- ( A. x ph -> A. x e. A ph )

Syntax hints:   -> wi 4   A.wal 1481   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

This theorem depends on definitions:  df-bi 197  df-ral 2917

This theorem is referenced by:  abnex  6965  find  7091  brdom5  9351  brdom4  9352  prodeq2w  14642  rpnnen2lem12  14954  elpotr  31686  phpreu  33393  neik0pk1imk0  38345  ordelordALTVD  39103  rexrsb  41169