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Theorem pm14.122a 27291
Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.122a  |-  ( A  e.  V  ->  ( A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem pm14.122a
StepHypRef Expression
1 albiim 1618 . 2  |-  ( A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  A. x ( x  =  A  ->  ph ) ) )
2 sbc6g 3129 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
32bicomd 193 . . 3  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  [. A  /  x ]. ph ) )
43anbi2d 685 . 2  |-  ( A  e.  V  ->  (
( A. x (
ph  ->  x  =  A )  /\  A. x
( x  =  A  ->  ph ) )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph ) ) )
51, 4syl5bb 249 1  |-  ( A  e.  V  ->  ( A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   [.wsbc 3104
This theorem is referenced by:  pm14.122c  27293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105
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