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Theorem cleqf 2923
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq 2718. See also cleqh 2855. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2365. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2129. (Revised by GG, 20-Aug-2023.)
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 dfcleq 2718 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1909 . . 3 𝑦(𝑥𝐴𝑥𝐵)
3 cleqf.1 . . . . 5 𝑥𝐴
43nfcri 2882 . . . 4 𝑥 𝑦𝐴
5 cleqf.2 . . . . 5 𝑥𝐵
65nfcri 2882 . . . 4 𝑥 𝑦𝐵
74, 6nfbi 1898 . . 3 𝑥(𝑦𝐴𝑦𝐵)
8 eleq1w 2808 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
9 eleq1w 2808 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
108, 9bibi12d 344 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
112, 7, 10cbvalv1 2331 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
121, 11bitr4i 277 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1531   = wceq 1533  wcel 2098  wnfc 2875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-cleq 2717  df-clel 2802  df-nfc 2877
This theorem is referenced by:  eqabf  2924  abid2fOLD  2926  eqvf  3471  eqrd  3996  eq0f  4340  iunab  5055  iinab  5072  mbfposr  25630  mbfinf  25643  itg1climres  25693  bnj1366  34593  bj-rabtrALT  36542  bj-rcleqf  36637  compab  44023  ssmapsn  44730  infnsuprnmpt  44766  pimrecltpos  46236  pimrecltneg  46252  smfaddlem1  46291  smflimsuplem7  46354  absnsb  46549
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