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Theorem bnj1538 28632
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1538.1  |-  A  =  { x  e.  B  |  ph }
Assertion
Ref Expression
bnj1538  |-  ( x  e.  A  ->  ph )

Proof of Theorem bnj1538
StepHypRef Expression
1 bnj1538.1 . . 3  |-  A  =  { x  e.  B  |  ph }
21rabeq2i 2861 . 2  |-  ( x  e.  A  <->  ( x  e.  B  /\  ph )
)
32simprbi 450 1  |-  ( x  e.  A  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   {crab 2623
This theorem is referenced by:  bnj1279  28793  bnj1311  28799  bnj1418  28815  bnj1312  28833  bnj1523  28846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-rab 2628
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