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Theorem eqvincf 2721
Description: A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
eqvincf.1  |-  F/_ x A
eqvincf.2  |-  F/_ x B
eqvincf.3  |-  A  e. 
_V
Assertion
Ref Expression
eqvincf  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )

Proof of Theorem eqvincf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqvincf.3 . . 3  |-  A  e. 
_V
21eqvinc 2719 . 2  |-  ( A  =  B  <->  E. y
( y  =  A  /\  y  =  B ) )
3 eqvincf.1 . . . . 5  |-  F/_ x A
43nfeq2 2231 . . . 4  |-  F/ x  y  =  A
5 eqvincf.2 . . . . 5  |-  F/_ x B
65nfeq2 2231 . . . 4  |-  F/ x  y  =  B
74, 6nfan 1498 . . 3  |-  F/ x
( y  =  A  /\  y  =  B )
8 nfv 1462 . . 3  |-  F/ y ( x  =  A  /\  x  =  B )
9 eqeq1 2088 . . . 4  |-  ( y  =  x  ->  (
y  =  A  <->  x  =  A ) )
10 eqeq1 2088 . . . 4  |-  ( y  =  x  ->  (
y  =  B  <->  x  =  B ) )
119, 10anbi12d 457 . . 3  |-  ( y  =  x  ->  (
( y  =  A  /\  y  =  B )  <->  ( x  =  A  /\  x  =  B ) ) )
127, 8, 11cbvex 1680 . 2  |-  ( E. y ( y  =  A  /\  y  =  B )  <->  E. x
( x  =  A  /\  x  =  B ) )
132, 12bitri 182 1  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   F/_wnfc 2207   _Vcvv 2602
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604
This theorem is referenced by: (None)
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