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Theorem cleqh 2134
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2198. (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 2031 . 2
2 ax-17 1416 . . . 4
3 dfbi2 368 . . . . 5
4 cleqh.1 . . . . . . 7
5 cleqh.2 . . . . . . 7
64, 5hbim 1434 . . . . . 6
75, 4hbim 1434 . . . . . 6
86, 7hban 1436 . . . . 5
93, 8hbxfrbi 1358 . . . 4
10 eleq1 2097 . . . . . 6
11 eleq1 2097 . . . . . 6
1210, 11bibi12d 224 . . . . 5
1312biimpd 132 . . . 4
142, 9, 13cbv3h 1628 . . 3
1512equcoms 1591 . . . . 5
1615biimprd 147 . . . 4
179, 2, 16cbv3h 1628 . . 3
1814, 17impbii 117 . 2
191, 18bitr4i 176 1
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242   wcel 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033
This theorem is referenced by:  abeq2  2143
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