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Theorem cleqh 2535
 Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (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 2432 . 2
2 ax-17 1627 . . . 4
3 dfbi2 611 . . . . 5
4 cleqh.1 . . . . . . 7
5 cleqh.2 . . . . . . 7
64, 5hbim 1837 . . . . . 6
75, 4hbim 1837 . . . . . 6
86, 7hban 1851 . . . . 5
93, 8hbxfrbi 1578 . . . 4
10 eleq1 2498 . . . . . 6
11 eleq1 2498 . . . . . 6
1210, 11bibi12d 314 . . . . 5
1312biimpd 200 . . . 4
142, 9, 13cbv3h 1973 . . 3
1512equcoms 1694 . . . . 5
1615biimprd 216 . . . 4
179, 2, 16cbv3h 1973 . . 3
1814, 17impbii 182 . 2
191, 18bitr4i 245 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wal 1550   wceq 1653   wcel 1726 This theorem is referenced by:  abeq2  2543  abeq2f  23962 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-cleq 2431  df-clel 2434
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