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Theorem nfel 2237
Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
nfnfc.1  |-  F/_ x A
nfeq.2  |-  F/_ x B
Assertion
Ref Expression
nfel  |-  F/ x  A  e.  B

Proof of Theorem nfel
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-clel 2084 . 2  |-  ( A  e.  B  <->  E. z
( z  =  A  /\  z  e.  B
) )
2 nfcv 2228 . . . . 5  |-  F/_ x
z
3 nfnfc.1 . . . . 5  |-  F/_ x A
42, 3nfeq 2236 . . . 4  |-  F/ x  z  =  A
5 nfeq.2 . . . . 5  |-  F/_ x B
65nfcri 2222 . . . 4  |-  F/ x  z  e.  B
74, 6nfan 1502 . . 3  |-  F/ x
( z  =  A  /\  z  e.  B
)
87nfex 1573 . 2  |-  F/ x E. z ( z  =  A  /\  z  e.  B )
91, 8nfxfr 1408 1  |-  F/ x  A  e.  B
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   F/wnf 1394   E.wex 1426    e. wcel 1438   F/_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  2758  elrabf  2769  sbcel12g  2946  nfdisjv  3834  rabxfrd  4291  ffnfvf  5457  elabgft1  11633  elabgf2  11635  bj-rspgt  11641
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