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Theorem cleqf 2217
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2153. (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 2050 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1437 . . 3 𝑦(𝑥𝐴𝑥𝐵)
3 cleqf.1 . . . . 5 𝑥𝐴
43nfcri 2188 . . . 4 𝑥 𝑦𝐴
5 cleqf.2 . . . . 5 𝑥𝐵
65nfcri 2188 . . . 4 𝑥 𝑦𝐵
74, 6nfbi 1497 . . 3 𝑥(𝑦𝐴𝑦𝐵)
8 eleq1 2116 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
9 eleq1 2116 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
108, 9bibi12d 228 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
112, 7, 10cbval 1653 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
121, 11bitr4i 180 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257   = wceq 1259  wcel 1409  wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183
This theorem is referenced by:  abid2f  2218  n0rf  3261  eq0  3267  iunab  3731  iinab  3746  sniota  4922
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