MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqvincf Structured version   Unicode version

Theorem eqvincf 3064
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 3063 . 2  |-  ( A  =  B  <->  E. y
( y  =  A  /\  y  =  B ) )
3 eqvincf.1 . . . . 5  |-  F/_ x A
43nfeq2 2583 . . . 4  |-  F/ x  y  =  A
5 eqvincf.2 . . . . 5  |-  F/_ x B
65nfeq2 2583 . . . 4  |-  F/ x  y  =  B
74, 6nfan 1846 . . 3  |-  F/ x
( y  =  A  /\  y  =  B )
8 nfv 1629 . . 3  |-  F/ y ( x  =  A  /\  x  =  B )
9 eqeq1 2442 . . . 4  |-  ( y  =  x  ->  (
y  =  A  <->  x  =  A ) )
10 eqeq1 2442 . . . 4  |-  ( y  =  x  ->  (
y  =  B  <->  x  =  B ) )
119, 10anbi12d 692 . . 3  |-  ( y  =  x  ->  (
( y  =  A  /\  y  =  B )  <->  ( x  =  A  /\  x  =  B ) ) )
127, 8, 11cbvex 1983 . 2  |-  ( E. y ( y  =  A  /\  y  =  B )  <->  E. x
( x  =  A  /\  x  =  B ) )
132, 12bitri 241 1  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   F/_wnfc 2559   _Vcvv 2956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958
  Copyright terms: Public domain W3C validator