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Theorem cleqh 2572
 Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2637. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2104. (Revised by BJ, 30-Nov-2020.)
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 2465 . 2
2 nfv 1769 . . 3
3 cleqh.1 . . . . 5
43nfi 1682 . . . 4
5 cleqh.2 . . . . 5
65nfi 1682 . . . 4
74, 6nfbi 2037 . . 3
8 eleq1 2537 . . . 4
9 eleq1 2537 . . . 4
108, 9bibi12d 328 . . 3
112, 7, 10cbvalv1 2083 . 2
121, 11bitr4i 260 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189  wal 1450   wceq 1452   wcel 1904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-cleq 2464  df-clel 2467 This theorem is referenced by:  abeq2  2580  abbi  2585  cleqf  2637  abeq2f  2639  bj-abeq2  31454
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