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Theorem pm14.122c 27024
Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.122c  |-  ( A  e.  V  ->  ( A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem pm14.122c
StepHypRef Expression
1 pm14.122a 27022 . 2  |-  ( A  e.  V  ->  ( A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph ) ) )
2 pm14.122b 27023 . 2  |-  ( A  e.  V  ->  (
( A. x (
ph  ->  x  =  A )  /\  [. A  /  x ]. ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
31, 2bitrd 246 1  |-  ( A  e.  V  ->  ( A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685   [.wsbc 2993
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-sbc 2994
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