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Theorem pm14.122b 26956
Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.122b  |-  ( A  e.  V  ->  (
( A. x (
ph  ->  x  =  A )  /\  [. A  /  x ]. ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem pm14.122b
StepHypRef Expression
1 eqeq2 2265 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21imbi2d 309 . . . . 5  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
32albidv 2005 . . . 4  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
4 dfsbcq 2937 . . . . 5  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
54bibi1d 312 . . . 4  |-  ( y  =  A  ->  (
( [. y  /  x ]. ph  <->  E. x ph )  <->  (
[. A  /  x ]. ph  <->  E. x ph )
) )
63, 5imbi12d 313 . . 3  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  ( [. y  /  x ]. ph  <->  E. x ph ) )  <->  ( A. x ( ph  ->  x  =  A )  -> 
( [. A  /  x ]. ph  <->  E. x ph )
) ) )
7 sbc5 2959 . . . 4  |-  ( [. y  /  x ]. ph  <->  E. x
( x  =  y  /\  ph ) )
8 nfa1 1719 . . . . 5  |-  F/ x A. x ( ph  ->  x  =  y )
9 simpr 449 . . . . . 6  |-  ( ( x  =  y  /\  ph )  ->  ph )
10 ancr 534 . . . . . . 7  |-  ( (
ph  ->  x  =  y )  ->  ( ph  ->  ( x  =  y  /\  ph ) ) )
1110a4s 1700 . . . . . 6  |-  ( A. x ( ph  ->  x  =  y )  -> 
( ph  ->  ( x  =  y  /\  ph ) ) )
129, 11impbid2 197 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( ( x  =  y  /\  ph )  <->  ph ) )
138, 12exbid 1714 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x ( x  =  y  /\  ph )  <->  E. x ph )
)
147, 13syl5bb 250 . . 3  |-  ( A. x ( ph  ->  x  =  y )  -> 
( [. y  /  x ]. ph  <->  E. x ph )
)
156, 14vtoclg 2794 . 2  |-  ( A  e.  V  ->  ( A. x ( ph  ->  x  =  A )  -> 
( [. A  /  x ]. ph  <->  E. x ph )
) )
1615pm5.32d 623 1  |-  ( A  e.  V  ->  (
( A. x (
ph  ->  x  =  A )  /\  [. A  /  x ]. ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1537    = 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
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