Theorem alral 1689

Description: Universal quantification implies restricted quantification.

Assertion

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

Proof of Theorem alral

Step	Hyp	Ref	Expression
1		ax-1 4	. . 3 |- (ph -> (x e. A -> ph))
2	1	19.20i 990	. 2 |- (A.xph -> A.x(x e. A -> ph))
3		df-ral 1646	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
4	2, 3	sylibr 200	1 |- (A.xph -> A.x e. A ph)

Syntax hints:   -> wi 3  A.wal 952   e. wcel 956  A.wral 1642

This theorem is referenced by:  brdom5 4782  brdom4 4783  gelcomplOLD 10353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973

This theorem depends on definitions:  df-bi 147  df-ral 1646

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