Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm14.122a Unicode version

Theorem pm14.122a 26955
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 1612 . 2  |-  ( A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  A. x ( x  =  A  ->  ph ) ) )
2 sbc6g 2960 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
32bicomd 194 . . 3  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  [. A  /  x ]. ph ) )
43anbi2d 687 . 2  |-  ( A  e.  V  ->  (
( A. x (
ph  ->  x  =  A )  /\  A. x
( x  =  A  ->  ph ) )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph ) ) )
51, 4syl5bb 250 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 6    <-> wb 178    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   [.wsbc 2935
This theorem is referenced by:  pm14.122c  26957
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-sbc 2936
  Copyright terms: Public domain W3C validator