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Theorem bnj228 28763
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
StepHypRef Expression
1 bnj228.1 . . 3  |-  ( ph  <->  A. x  e.  A  ps )
2 rsp 2603 . . 3  |-  ( A. x  e.  A  ps  ->  ( x  e.  A  ->  ps ) )
31, 2sylbi 187 . 2  |-  ( ph  ->  ( x  e.  A  ->  ps ) )
43impcom 419 1  |-  ( ( x  e.  A  /\  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543
This theorem is referenced by:  bnj229  28916  bnj999  28989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ral 2548
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