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Theorem nfel 2291
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 2136 . 2 (𝐴𝐵 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝐵))
2 nfcv 2282 . . . . 5 𝑥𝑧
3 nfnfc.1 . . . . 5 𝑥𝐴
42, 3nfeq 2290 . . . 4 𝑥 𝑧 = 𝐴
5 nfeq.2 . . . . 5 𝑥𝐵
65nfcri 2276 . . . 4 𝑥 𝑧𝐵
74, 6nfan 1545 . . 3 𝑥(𝑧 = 𝐴𝑧𝐵)
87nfex 1617 . 2 𝑥𝑧(𝑧 = 𝐴𝑧𝐵)
91, 8nfxfr 1451 1 𝑥 𝐴𝐵
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1332  wnf 1437  wex 1469  wcel 1481  wnfc 2269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271
This theorem is referenced by:  nfel1  2293  nfel2  2295  nfnel  2411  elabgf  2830  elrabf  2842  sbcel12g  3022  nfdisjv  3926  rabxfrd  4398  ffnfvf  5587  mptelixpg  6636  elabgft1  13156  elabgf2  13158  bj-rspgt  13164
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