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Theorem cleqf 2786
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2721. (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 2754 . 2 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 cleqf.2 . . 3 𝑥𝐵
43nfcrii 2754 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
52, 4cleqh 2721 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1478   = wceq 1480  wcel 1987  wnfc 2748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-cleq 2614  df-clel 2617  df-nfc 2750
This theorem is referenced by:  abid2f  2787  eqvf  3190  eq0f  3901  n0fOLD  3904  iunab  4532  iinab  4547  mbfposr  23325  mbfinf  23338  itg1climres  23387  bnj1366  30605  bj-rabtrALT  32571  compab  38124  dfcleqf  38737
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