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Theorem bnj1454 28544
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1454.1  |-  A  =  { x  |  ph }
Assertion
Ref Expression
bnj1454  |-  ( B  e.  _V  ->  ( B  e.  A  <->  [. B  /  x ]. ph ) )

Proof of Theorem bnj1454
StepHypRef Expression
1 df-sbc 3098 . . 3  |-  ( [. B  /  x ]. ph  <->  B  e.  { x  |  ph }
)
21a1i 11 . 2  |-  ( B  e.  _V  ->  ( [. B  /  x ]. ph  <->  B  e.  { x  |  ph } ) )
3 bnj1454.1 . . 3  |-  A  =  { x  |  ph }
43eleq2i 2444 . 2  |-  ( B  e.  A  <->  B  e.  { x  |  ph }
)
52, 4syl6rbbr 256 1  |-  ( B  e.  _V  ->  ( B  e.  A  <->  [. B  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   {cab 2366   _Vcvv 2892   [.wsbc 3097
This theorem is referenced by:  bnj1452  28752  bnj1463  28755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-cleq 2373  df-clel 2376  df-sbc 3098
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