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Theorem bnj1465 29193
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 . . . . 5  |-  ( ch 
->  ps )
21adantr 451 . . . 4  |-  ( ( ch  /\  A  e.  V )  ->  ps )
3 bnj1465.2 . . . . . 6  |-  ( ps 
->  A. x ps )
4 bnj1465.1 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
53, 4bnj1464 29192 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
65adantl 452 . . . 4  |-  ( ( ch  /\  A  e.  V )  ->  ( [. A  /  x ]. ph  <->  ps ) )
72, 6mpbird 223 . . 3  |-  ( ( ch  /\  A  e.  V )  ->  [. A  /  x ]. ph )
8 sbc5 3028 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
97, 8sylib 188 . 2  |-  ( ( ch  /\  A  e.  V )  ->  E. x
( x  =  A  /\  ph ) )
109bnj1266 29160 1  |-  ( ( ch  /\  A  e.  V )  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  bnj1463  29401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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