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Theorem cleqf 2933
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2867. (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 2900 . 2 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 cleqf.2 . . 3 𝑥𝐵
43nfcrii 2900 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
52, 4cleqh 2867 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1650   = wceq 1652  wcel 2155  wnfc 2894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-cleq 2758  df-clel 2761  df-nfc 2896
This theorem is referenced by:  abid2f  2934  eqvf  3357  eqrd  3780  eq0f  4090  iunab  4722  iinab  4737  mbfposr  23710  mbfinf  23723  itg1climres  23772  bnj1366  31280  bj-rabtrALT  33287  compab  39250  dfcleqf  39838  absnsb  41741
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