Theorem bnj1459 30913

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

Hypotheses

Ref	Expression
bnj1459.1	|- ( ps <-> ( ph /\ x e. A ) )
bnj1459.2	|- ( ps -> ch )

Assertion

Ref	Expression
bnj1459	|- ( ph -> A. x e. A ch )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem bnj1459

Step	Hyp	Ref	Expression
1		bnj1459.1	. . 3 |- ( ps <-> ( ph /\ x e. A ) )
2		bnj1459.2	. . 3 |- ( ps -> ch )
3	1, 2	sylbir 225	. 2 |- ( ( ph /\ x e. A ) -> ch )
4	3	ralrimiva 2966	1 |- ( ph -> A. x e. A ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   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  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917

This theorem is referenced by:  bnj1501  31135