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Theorem cleqf 2513
 Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (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 2347 . 2 (A = By(y Ay B))
2 nfv 1619 . . 3 y(x Ax B)
3 cleqf.1 . . . . 5 xA
43nfcri 2483 . . . 4 x y A
5 cleqf.2 . . . . 5 xB
65nfcri 2483 . . . 4 x y B
74, 6nfbi 1834 . . 3 x(y Ay B)
8 eleq1 2413 . . . 4 (x = y → (x Ay A))
9 eleq1 2413 . . . 4 (x = y → (x By B))
108, 9bibi12d 312 . . 3 (x = y → ((x Ax B) ↔ (y Ay B)))
112, 7, 10cbval 1984 . 2 (x(x Ax B) ↔ y(y Ay B))
121, 11bitr4i 243 1 (A = Bx(x Ax B))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478 This theorem is referenced by:  abid2f  2514  n0f  3558  iunab  4012  iinab  4027  sniota  4369
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