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Theorem bnj1454 34320
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 3325 . . 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 2532 . 2  |-  ( B  e.  A  <->  B  e.  { x  |  ph }
)
52, 4syl6rbbr 264 1  |-  ( B  e.  _V  ->  ( B  e.  A  <->  [. B  /  x ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   {cab 2439   _Vcvv 3106   [.wsbc 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-cleq 2446  df-clel 2449  df-sbc 3325
This theorem is referenced by:  bnj1452  34528  bnj1463  34531
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