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Theorem bnj1465 29216
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 452 . . 3  |-  ( ( ch  /\  A  e.  V )  ->  ps )
3 bnj1465.2 . . . . 5  |-  ( ps 
->  A. x ps )
4 bnj1465.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
53, 4bnj1464 29215 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
65adantl 453 . . 3  |-  ( ( ch  /\  A  e.  V )  ->  ( [. A  /  x ]. ph  <->  ps ) )
72, 6mpbird 224 . 2  |-  ( ( ch  /\  A  e.  V )  ->  [. A  /  x ]. ph )
87spesbcd 3243 1  |-  ( ( ch  /\  A  e.  V )  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   [.wsbc 3161
This theorem is referenced by:  bnj1463  29424
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-sbc 3162
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