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Theorem eqvincf 2896
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 2895 . 2  |-  ( A  =  B  <->  E. y
( y  =  A  /\  y  =  B ) )
3 eqvincf.1 . . . . 5  |-  F/_ x A
43nfeq2 2430 . . . 4  |-  F/ x  y  =  A
5 eqvincf.2 . . . . 5  |-  F/_ x B
65nfeq2 2430 . . . 4  |-  F/ x  y  =  B
74, 6nfan 1771 . . 3  |-  F/ x
( y  =  A  /\  y  =  B )
8 nfv 1605 . . 3  |-  F/ y ( x  =  A  /\  x  =  B )
9 eqeq1 2289 . . . 4  |-  ( y  =  x  ->  (
y  =  A  <->  x  =  A ) )
10 eqeq1 2289 . . . 4  |-  ( y  =  x  ->  (
y  =  B  <->  x  =  B ) )
119, 10anbi12d 691 . . 3  |-  ( y  =  x  ->  (
( y  =  A  /\  y  =  B )  <->  ( x  =  A  /\  x  =  B ) ) )
127, 8, 11cbvex 1925 . 2  |-  ( E. y ( y  =  A  /\  y  =  B )  <->  E. x
( x  =  A  /\  x  =  B ) )
132, 12bitri 240 1  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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