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Theorem cleqh 2115
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2179. (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 2012 . 2
2 ax-17 1396 . . . 4
3 dfbi2 368 . . . . 5
4 cleqh.1 . . . . . . 7
5 cleqh.2 . . . . . . 7
64, 5hbim 1415 . . . . . 6
75, 4hbim 1415 . . . . . 6
86, 7hban 1417 . . . . 5
93, 8hbxfrbi 1337 . . . 4
10 eleq1 2078 . . . . . 6
11 eleq1 2078 . . . . . 6
1210, 11bibi12d 224 . . . . 5
1312biimpd 132 . . . 4
142, 9, 13cbv3h 1609 . . 3
1512equcoms 1572 . . . . 5
1615biimprd 147 . . . 4
179, 2, 16cbv3h 1609 . . 3
1814, 17impbii 117 . 2
191, 18bitr4i 176 1
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1224   wceq 1226   wcel 1370
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-cleq 2011  df-clel 2014
This theorem is referenced by:  abeq2  2124
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