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Theorem eqvincf 3194
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 3193 . 2  |-  ( A  =  B  <->  E. y
( y  =  A  /\  y  =  B ) )
3 eqvincf.1 . . . . 5  |-  F/_ x A
43nfeq2 2633 . . . 4  |-  F/ x  y  =  A
5 eqvincf.2 . . . . 5  |-  F/_ x B
65nfeq2 2633 . . . 4  |-  F/ x  y  =  B
74, 6nfan 1866 . . 3  |-  F/ x
( y  =  A  /\  y  =  B )
8 nfv 1674 . . 3  |-  F/ y ( x  =  A  /\  x  =  B )
9 eqeq1 2458 . . . 4  |-  ( y  =  x  ->  (
y  =  A  <->  x  =  A ) )
10 eqeq1 2458 . . . 4  |-  ( y  =  x  ->  (
y  =  B  <->  x  =  B ) )
119, 10anbi12d 710 . . 3  |-  ( y  =  x  ->  (
( y  =  A  /\  y  =  B )  <->  ( x  =  A  /\  x  =  B ) ) )
127, 8, 11cbvex 1982 . 2  |-  ( E. y ( y  =  A  /\  y  =  B )  <->  E. x
( x  =  A  /\  x  =  B ) )
132, 12bitri 249 1  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   F/_wnfc 2602   _Vcvv 3078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080
This theorem is referenced by: (None)
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