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Theorem ceqex 2742
Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
Assertion
Ref Expression
ceqex  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem ceqex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 19.8a 1527 . . 3  |-  ( x  =  A  ->  E. x  x  =  A )
2 isset 2625 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
31, 2sylibr 132 . 2  |-  ( x  =  A  ->  A  e.  _V )
4 eqeq2 2097 . . . 4  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
54anbi1d 453 . . . . . 6  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
65exbidv 1753 . . . . 5  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
76bibi2d 230 . . . 4  |-  ( y  =  A  ->  (
( ph  <->  E. x ( x  =  y  /\  ph ) )  <->  ( ph  <->  E. x ( x  =  A  /\  ph )
) ) )
84, 7imbi12d 232 . . 3  |-  ( y  =  A  ->  (
( x  =  y  ->  ( ph  <->  E. x
( x  =  y  /\  ph ) ) )  <->  ( x  =  A  ->  ( ph  <->  E. x ( x  =  A  /\  ph )
) ) ) )
9 19.8a 1527 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  E. x
( x  =  y  /\  ph ) )
109ex 113 . . . 4  |-  ( x  =  y  ->  ( ph  ->  E. x ( x  =  y  /\  ph ) ) )
11 vex 2622 . . . . . 6  |-  y  e. 
_V
1211alexeq 2741 . . . . 5  |-  ( A. x ( x  =  y  ->  ph )  <->  E. x
( x  =  y  /\  ph ) )
13 sp 1446 . . . . . 6  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
1413com12 30 . . . . 5  |-  ( x  =  y  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
1512, 14syl5bir 151 . . . 4  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) )
1610, 15impbid 127 . . 3  |-  ( x  =  y  ->  ( ph 
<->  E. x ( x  =  y  /\  ph ) ) )
178, 16vtoclg 2679 . 2  |-  ( A  e.  _V  ->  (
x  =  A  -> 
( ph  <->  E. x ( x  =  A  /\  ph ) ) ) )
183, 17mpcom 36 1  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  ceqsexg  2743  sbc6g  2862
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