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Theorem nfel 2308
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 2153 . 2  |-  ( A  e.  B  <->  E. z
( z  =  A  /\  z  e.  B
) )
2 nfcv 2299 . . . . 5  |-  F/_ x
z
3 nfnfc.1 . . . . 5  |-  F/_ x A
42, 3nfeq 2307 . . . 4  |-  F/ x  z  =  A
5 nfeq.2 . . . . 5  |-  F/_ x B
65nfcri 2293 . . . 4  |-  F/ x  z  e.  B
74, 6nfan 1545 . . 3  |-  F/ x
( z  =  A  /\  z  e.  B
)
87nfex 1617 . 2  |-  F/ x E. z ( z  =  A  /\  z  e.  B )
91, 8nfxfr 1454 1  |-  F/ x  A  e.  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1335   F/wnf 1440   E.wex 1472    e. wcel 2128   F/_wnfc 2286
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288
This theorem is referenced by:  nfel1  2310  nfel2  2312  nfnel  2429  elabgf  2854  elrabf  2866  sbcel12g  3046  nfdisjv  3955  rabxfrd  4430  ffnfvf  5627  mptelixpg  6680  elabgft1  13394  elabgf2  13396  bj-rspgt  13402
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