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Theorem bnj525 34195
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj525.1  |-  A  e. 
_V
Assertion
Ref Expression
bnj525  |-  ( [. A  /  x ]. ph  <->  ph )
Distinct variable group:    ph, x
Allowed substitution hint:    A( x)

Proof of Theorem bnj525
StepHypRef Expression
1 bnj525.1 . 2  |-  A  e. 
_V
2 sbcg 3390 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  ph ) )
31, 2ax-mp 5 1  |-  ( [. A  /  x ]. ph  <->  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1823   _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-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108  df-sbc 3325
This theorem is referenced by:  bnj538OLD  34198  bnj976  34237  bnj91  34320  bnj92  34321  bnj523  34346  bnj539  34350  bnj540  34351  bnj1040  34429
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