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Theorem nfel 2497
 Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
nfnfc.1 xA
nfeq.2 xB
Assertion
Ref Expression
nfel x A B

Proof of Theorem nfel
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-clel 2349 . 2 (A Bz(z = A z B))
2 nfcv 2489 . . . . 5 xz
3 nfnfc.1 . . . . 5 xA
42, 3nfeq 2496 . . . 4 x z = A
5 nfeq.2 . . . . 5 xB
65nfcri 2483 . . . 4 x z B
74, 6nfan 1824 . . 3 x(z = A z B)
87nfex 1843 . 2 xz(z = A z B)
91, 8nfxfr 1570 1 x A B
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = 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:  nfel1  2499  nfel2  2501  nfnel  2611  elabgf  2983  elrabf  2993  sbcel12g  3151  ffnfvf  5428
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