ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cleqf Unicode version

Theorem cleqf 2306
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2240. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
cleqf.1  |-  F/_ x A
cleqf.2  |-  F/_ x B
Assertion
Ref Expression
cleqf  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )

Proof of Theorem cleqf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2134 . 2  |-  ( A  =  B  <->  A. y
( y  e.  A  <->  y  e.  B ) )
2 nfv 1509 . . 3  |-  F/ y ( x  e.  A  <->  x  e.  B )
3 cleqf.1 . . . . 5  |-  F/_ x A
43nfcri 2276 . . . 4  |-  F/ x  y  e.  A
5 cleqf.2 . . . . 5  |-  F/_ x B
65nfcri 2276 . . . 4  |-  F/ x  y  e.  B
74, 6nfbi 1569 . . 3  |-  F/ x
( y  e.  A  <->  y  e.  B )
8 eleq1 2203 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
9 eleq1 2203 . . . 4  |-  ( x  =  y  ->  (
x  e.  B  <->  y  e.  B ) )
108, 9bibi12d 234 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  <->  x  e.  B )  <->  ( y  e.  A  <->  y  e.  B
) ) )
112, 7, 10cbval 1728 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  B
)  <->  A. y ( y  e.  A  <->  y  e.  B ) )
121, 11bitr4i 186 1  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1330    = wceq 1332    e. wcel 1481   F/_wnfc 2269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271
This theorem is referenced by:  abid2f  2307  n0rf  3379  eq0  3385  iunab  3866  iinab  3881  sniota  5122
  Copyright terms: Public domain W3C validator