Theorem bnj228 30803

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj228.1	|- ( ph <-> A. x e. A ps )

Assertion

Ref	Expression
bnj228	|- ( ( x e. A /\ ph ) -> ps )

Proof of Theorem bnj228

Step	Hyp	Ref	Expression
1		bnj228.1	. . 3 |- ( ph <-> A. x e. A ps )
2		rsp 2929	. . 3 |- ( A. x e. A ps -> ( x e. A -> ps ) )
3	1, 2	sylbi 207	. 2 |- ( ph -> ( x e. A -> ps ) )
4	3	impcom 446	1 |- ( ( x e. A /\ ph ) -> ps )

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  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917

This theorem is referenced by:  bnj229  30954  bnj999  31027