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Theorem cleqf 2184
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2120. (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 = Bx(x Ax B))

Proof of Theorem cleqf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2017 . 2 (A = By(y Ay B))
2 nfv 1403 . . 3 y(x Ax B)
3 cleqf.1 . . . . 5 xA
43nfcri 2155 . . . 4 x y A
5 cleqf.2 . . . . 5 xB
65nfcri 2155 . . . 4 x y B
74, 6nfbi 1465 . . 3 x(y Ay B)
8 eleq1 2083 . . . 4 (x = y → (x Ay A))
9 eleq1 2083 . . . 4 (x = y → (x By B))
108, 9bibi12d 224 . . 3 (x = y → ((x Ax B) ↔ (y Ay B)))
112, 7, 10cbval 1620 . 2 (x(x Ax B) ↔ y(y Ay B))
121, 11bitr4i 176 1 (A = Bx(x Ax B))
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1226   = wceq 1228   wcel 1375  wnfc 2148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-cleq 2016  df-clel 2019  df-nfc 2150
This theorem is referenced by:  abid2f  2185  n0rf  3209  eq0  3215  iunab  3676  iinab  3691  sniota  4819
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