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Theorem cleqf 2259
 Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2194. (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 2089 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1473 . . 3 𝑦(𝑥𝐴𝑥𝐵)
3 cleqf.1 . . . . 5 𝑥𝐴
43nfcri 2229 . . . 4 𝑥 𝑦𝐴
5 cleqf.2 . . . . 5 𝑥𝐵
65nfcri 2229 . . . 4 𝑥 𝑦𝐵
74, 6nfbi 1533 . . 3 𝑥(𝑦𝐴𝑦𝐵)
8 eleq1 2157 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
9 eleq1 2157 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
108, 9bibi12d 234 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
112, 7, 10cbval 1691 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
121, 11bitr4i 186 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  ∀wal 1294   = wceq 1296   ∈ wcel 1445  Ⅎwnfc 2222 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-cleq 2088  df-clel 2091  df-nfc 2224 This theorem is referenced by:  abid2f  2260  n0rf  3314  eq0  3320  iunab  3798  iinab  3813  sniota  5041
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