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Theorem bnj1454 29150
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 3154 . . 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 2499 . 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 1652    e. wcel 1725   {cab 2421   _Vcvv 2948   [.wsbc 3153
This theorem is referenced by:  bnj1452  29358  bnj1463  29361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2428  df-clel 2431  df-sbc 3154
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