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Theorem cleqh 2450
 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 2347 . 2
2 ax-17 1616 . . . 4
3 dfbi2 609 . . . . 5
4 cleqh.1 . . . . . . 7
5 cleqh.2 . . . . . . 7
64, 5hbim 1817 . . . . . 6
75, 4hbim 1817 . . . . . 6
86, 7hban 1828 . . . . 5
93, 8hbxfrbi 1568 . . . 4
10 eleq1 2413 . . . . . 6
11 eleq1 2413 . . . . . 6
1210, 11bibi12d 312 . . . . 5
1312biimpd 198 . . . 4
142, 9, 13cbv3h 1983 . . 3
1512equcoms 1681 . . . . 5
1615biimprd 214 . . . 4
179, 2, 16cbv3h 1983 . . 3
1814, 17impbii 180 . 2
191, 18bitr4i 243 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wal 1540   wceq 1642   wcel 1710 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349 This theorem is referenced by:  abeq2  2458
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