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Theorem cleqh 2179
 Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2243. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
cleqh.1
cleqh.2
Assertion
Ref Expression
cleqh
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem cleqh
StepHypRef Expression
1 dfcleq 2076 . 2
2 ax-17 1460 . . . 4
3 dfbi2 380 . . . . 5
4 cleqh.1 . . . . . . 7
5 cleqh.2 . . . . . . 7
64, 5hbim 1478 . . . . . 6
75, 4hbim 1478 . . . . . 6
86, 7hban 1480 . . . . 5
93, 8hbxfrbi 1402 . . . 4
10 eleq1 2142 . . . . . 6
11 eleq1 2142 . . . . . 6
1210, 11bibi12d 233 . . . . 5
1312biimpd 142 . . . 4
142, 9, 13cbv3h 1672 . . 3
1512equcoms 1635 . . . . 5
1615biimprd 156 . . . 4
179, 2, 16cbv3h 1672 . . 3
1814, 17impbii 124 . 2
191, 18bitr4i 185 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103  wal 1283   wceq 1285   wcel 1434 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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  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-nf 1391  df-cleq 2075  df-clel 2078 This theorem is referenced by:  abeq2  2188
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