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Theorem pm14.122a 27633
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 1600 . 2  |-  ( A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  A. x ( x  =  A  ->  ph ) ) )
2 sbc6g 3018 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
32bicomd 192 . . 3  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  [. A  /  x ]. ph ) )
43anbi2d 684 . 2  |-  ( A  e.  V  ->  (
( A. x (
ph  ->  x  =  A )  /\  A. x
( x  =  A  ->  ph ) )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph ) ) )
51, 4syl5bb 248 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 176    /\ wa 358   A.wal 1529    = wceq 1625    e. wcel 1686   [.wsbc 2993
This theorem is referenced by:  pm14.122c  27635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  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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