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Theorem cleqf 2337
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2270. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
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 2164 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1521 . . 3 𝑦(𝑥𝐴𝑥𝐵)
3 cleqf.1 . . . . 5 𝑥𝐴
43nfcri 2306 . . . 4 𝑥 𝑦𝐴
5 cleqf.2 . . . . 5 𝑥𝐵
65nfcri 2306 . . . 4 𝑥 𝑦𝐵
74, 6nfbi 1582 . . 3 𝑥(𝑦𝐴𝑦𝐵)
8 eleq1 2233 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
9 eleq1 2233 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
108, 9bibi12d 234 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
112, 7, 10cbval 1747 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
121, 11bitr4i 186 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1346   = wceq 1348  wcel 2141  wnfc 2299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301
This theorem is referenced by:  abid2f  2338  n0rf  3427  eq0  3433  iunab  3919  iinab  3934  sniota  5189
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