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Theorem cleqf 2776
 Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2711. (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 2744 . 2 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 cleqf.2 . . 3 𝑥𝐵
43nfcrii 2744 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
52, 4cleqh 2711 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740 This theorem is referenced by:  abid2f  2777  eqvf  3177  eq0f  3884  n0fOLD  3887  iunab  4502  iinab  4517  mbfposr  23225  mbfinf  23238  itg1climres  23287  bnj1366  30154  bj-rabtrALT  32119  compab  37666
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