Theorem alral 2595

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 1508	. 2 |- ( A. x ph -> A. x ( x e. A -> ph ) )
3		df-ral 2533	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4	2, 3	sylibr 134	1 |- ( A. x ph -> A. x e. A ph )

This proof depends on syntax axioms:   -> wi 4   A.wal 1400   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-ia3 108  ax-5 1500  ax-gen 1502

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

This theorem is used by:  abnex  4593  find  4746  prodeq2w  12323  findset  16971