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Theorem nfel 2168
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 2018 . 2 (A Bz(z = A z B))
2 nfcv 2160 . . . . 5 xz
3 nfnfc.1 . . . . 5 xA
42, 3nfeq 2167 . . . 4 x z = A
5 nfeq.2 . . . . 5 xB
65nfcri 2154 . . . 4 x z B
74, 6nfan 1439 . . 3 x(z = A z B)
87nfex 1510 . 2 xz(z = A z B)
91, 8nfxfr 1343 1 x A B
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wnf 1329  wex 1362   wcel 1374  wnfc 2147
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-cleq 2015  df-clel 2018  df-nfc 2149
This theorem is referenced by:  nfel1  2170  nfel2  2172  nfnel  2282  elabgf  2662  elrabf  2673  sbcel12g  2842  nfdisjv  3731  rabxfrd  4151  ffnfvf  5249  elabgft1  7171  elabgf2  7173  bj-rspgt  7179
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