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Theorem cleqh 2572
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2646. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.)
Hypotheses
Ref Expression
cleqh.1
cleqh.2
Assertion
Ref Expression
cleqh
Distinct variable groups:   ,   ,   ,

Proof of Theorem cleqh
StepHypRef Expression
1 dfcleq 2450 . 2
2 nfv 1707 . . 3
3 cleqh.1 . . . . 5
43nfi 1623 . . . 4
5 cleqh.2 . . . . 5
65nfi 1623 . . . 4
74, 6nfbi 1934 . . 3
8 eleq1 2529 . . . 4
9 eleq1 2529 . . . 4
108, 9bibi12d 321 . . 3
112, 7, 10cbval 2021 . 2
121, 11bitr4i 252 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  e.wcel 1818
This theorem is referenced by:  abeq2  2581  abbi  2588  cleqf  2646  abeq2f  27398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-cleq 2449  df-clel 2452
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