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Theorem nfel 2186
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 2036 . 2 (𝐴𝐵 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝐵))
2 nfcv 2178 . . . . 5 𝑥𝑧
3 nfnfc.1 . . . . 5 𝑥𝐴
42, 3nfeq 2185 . . . 4 𝑥 𝑧 = 𝐴
5 nfeq.2 . . . . 5 𝑥𝐵
65nfcri 2172 . . . 4 𝑥 𝑧𝐵
74, 6nfan 1457 . . 3 𝑥(𝑧 = 𝐴𝑧𝐵)
87nfex 1528 . 2 𝑥𝑧(𝑧 = 𝐴𝑧𝐵)
91, 8nfxfr 1363 1 𝑥 𝐴𝐵
Colors of variables: wff set class
Syntax hints:  wa 97   = wceq 1243  wnf 1349  wex 1381  wcel 1393  wnfc 2165
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167
This theorem is referenced by:  nfel1  2188  nfel2  2190  nfnel  2304  elabgf  2685  elrabf  2696  sbcel12g  2865  nfdisjv  3757  rabxfrd  4201  ffnfvf  5324  elabgft1  9917  elabgf2  9919  bj-rspgt  9925
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