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Theorem cleqf 2305
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2239. (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 2133 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1508 . . 3 𝑦(𝑥𝐴𝑥𝐵)
3 cleqf.1 . . . . 5 𝑥𝐴
43nfcri 2275 . . . 4 𝑥 𝑦𝐴
5 cleqf.2 . . . . 5 𝑥𝐵
65nfcri 2275 . . . 4 𝑥 𝑦𝐵
74, 6nfbi 1568 . . 3 𝑥(𝑦𝐴𝑦𝐵)
8 eleq1 2202 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
9 eleq1 2202 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
108, 9bibi12d 234 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
112, 7, 10cbval 1727 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
121, 11bitr4i 186 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1329   = wceq 1331  wcel 1480  wnfc 2268
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270
This theorem is referenced by:  abid2f  2306  n0rf  3375  eq0  3381  iunab  3859  iinab  3874  sniota  5115
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