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Theorem nfel 2183
 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 2033 . 2 (A Bz(z = A z B))
2 nfcv 2175 . . . . 5 xz
3 nfnfc.1 . . . . 5 xA
42, 3nfeq 2182 . . . 4 x z = A
5 nfeq.2 . . . . 5 xB
65nfcri 2169 . . . 4 x z B
74, 6nfan 1454 . . 3 x(z = A z B)
87nfex 1525 . 2 xz(z = A z B)
91, 8nfxfr 1360 1 x A B
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  Ⅎwnf 1346  ∃wex 1378   ∈ wcel 1390  Ⅎwnfc 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:  nfel1  2185  nfel2  2187  nfnel  2298  elabgf  2679  elrabf  2690  sbcel12g  2859  nfdisjv  3747  rabxfrd  4166  ffnfvf  5265  elabgft1  8851  elabgf2  8853  bj-rspgt  8859
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