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Theorem cleqf 2939
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2873. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
Hypotheses
Ref Expression
cleqf.1 𝑥𝐴
cleqf.2 𝑥𝐵
Assertion
Ref Expression
cleqf (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Proof of Theorem cleqf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cleqf.1 . . 3 𝑥𝐴
21nfcrii 2906 . 2 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 cleqf.2 . . 3 𝑥𝐵
43nfcrii 2906 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
52, 4cleqh 2873 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1629   = wceq 1631  wcel 2145  wnfc 2900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-cleq 2764  df-clel 2767  df-nfc 2902
This theorem is referenced by:  abid2f  2940  eqvf  3355  eqrd  3771  eq0f  4073  iunab  4700  iinab  4715  mbfposr  23639  mbfinf  23652  itg1climres  23701  bnj1366  31238  bj-rabtrALT  33259  compab  39171  dfcleqf  39776
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