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Theorem bnj1465 32190
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1465.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
bnj1465.2  |-  ( ps 
->  A. x ps )
bnj1465.3  |-  ( ch 
->  ps )
Assertion
Ref Expression
bnj1465  |-  ( ( ch  /\  A  e.  V )  ->  E. x ph )
Distinct variable groups:    x, A    x, V
Allowed substitution hints:    ph( x)    ps( x)    ch( x)

Proof of Theorem bnj1465
StepHypRef Expression
1 bnj1465.3 . . . 4  |-  ( ch 
->  ps )
21adantr 465 . . 3  |-  ( ( ch  /\  A  e.  V )  ->  ps )
3 bnj1465.2 . . . . 5  |-  ( ps 
->  A. x ps )
4 bnj1465.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
53, 4bnj1464 32189 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
65adantl 466 . . 3  |-  ( ( ch  /\  A  e.  V )  ->  ( [. A  /  x ]. ph  <->  ps ) )
72, 6mpbird 232 . 2  |-  ( ( ch  /\  A  e.  V )  ->  [. A  /  x ]. ph )
87spesbcd 3388 1  |-  ( ( ch  /\  A  e.  V )  ->  E. x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   [.wsbc 3294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-sbc 3295
This theorem is referenced by:  bnj1463  32398
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