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Theorem nfel 2237
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 𝐴𝐵

Proof of Theorem nfel
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-clel 2084 . 2 (𝐴𝐵 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝐵))
2 nfcv 2228 . . . . 5 𝑥𝑧
3 nfnfc.1 . . . . 5 𝑥𝐴
42, 3nfeq 2236 . . . 4 𝑥 𝑧 = 𝐴
5 nfeq.2 . . . . 5 𝑥𝐵
65nfcri 2222 . . . 4 𝑥 𝑧𝐵
74, 6nfan 1502 . . 3 𝑥(𝑧 = 𝐴𝑧𝐵)
87nfex 1573 . 2 𝑥𝑧(𝑧 = 𝐴𝑧𝐵)
91, 8nfxfr 1408 1 𝑥 𝐴𝐵
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wnf 1394  wex 1426  wcel 1438  wnfc 2215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217
This theorem is referenced by:  nfel1  2239  nfel2  2241  nfnel  2357  elabgf  2756  elrabf  2767  sbcel12g  2944  nfdisjv  3826  rabxfrd  4282  ffnfvf  5441  elabgft1  11335  elabgf2  11337  bj-rspgt  11343
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