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Theorem cleqf 2198
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2134. (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 2031 . 2 (A = By(y Ay B))
2 nfv 1418 . . 3 y(x Ax B)
3 cleqf.1 . . . . 5 xA
43nfcri 2169 . . . 4 x y A
5 cleqf.2 . . . . 5 xB
65nfcri 2169 . . . 4 x y B
74, 6nfbi 1478 . . 3 x(y Ay B)
8 eleq1 2097 . . . 4 (x = y → (x Ay A))
9 eleq1 2097 . . . 4 (x = y → (x By B))
108, 9bibi12d 224 . . 3 (x = y → ((x Ax B) ↔ (y Ay B)))
112, 7, 10cbval 1634 . 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 1240   = wceq 1242   wcel 1390  wnfc 2162
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164
This theorem is referenced by:  abid2f  2199  n0rf  3227  eq0  3233  iunab  3694  iinab  3709  sniota  4837
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