MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cleqf Structured version   Visualization version   GIF version

Theorem cleqf 3010
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions as in dfcleq 2815. See also cleqh 2936. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2386. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2141. (Revised by Gino Giotto, 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 2815 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1911 . . 3 𝑦(𝑥𝐴𝑥𝐵)
3 cleqf.1 . . . . 5 𝑥𝐴
43nfcriv 2967 . . . 4 𝑥 𝑦𝐴
5 cleqf.2 . . . . 5 𝑥𝐵
65nfcriv 2967 . . . 4 𝑥 𝑦𝐵
74, 6nfbi 1900 . . 3 𝑥(𝑦𝐴𝑦𝐵)
8 eleq1w 2895 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
9 eleq1w 2895 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
108, 9bibi12d 348 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
112, 7, 10cbvalv1 2357 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
121, 11bitr4i 280 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wal 1531   = wceq 1533  wcel 2110  wnfc 2961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-nfc 2963
This theorem is referenced by:  abid2f  3012  abeq2f  3013  eqvf  3504  eqrd  3986  eq0f  4305  iunab  4968  iinab  4983  mbfposr  24247  mbfinf  24260  itg1climres  24309  bnj1366  32096  bj-rabtrALT  34244  bj-rcleqf  34332  compab  40767  ssmapsn  41471  infnsuprnmpt  41514  pimrecltpos  42980  pimrecltneg  42994  smfaddlem1  43032  smflimsuplem7  43093  absnsb  43255
  Copyright terms: Public domain W3C validator