Theorem bnj1142 30860

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1142.1	|- ( ph -> A. x ( x e. A -> ps ) )

Assertion

Ref	Expression
bnj1142	|- ( ph -> A. x e. A ps )

Proof of Theorem bnj1142

Step	Hyp	Ref	Expression
1		bnj1142.1	. 2 |- ( ph -> A. x ( x e. A -> ps ) )
2		df-ral 2917	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
3	1, 2	sylibr 224	1 |- ( ph -> A. x e. A ps )

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

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

This theorem is referenced by:  bnj1476  30917  bnj1533  30922  bnj1523  31139