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Theorem nfel 2321
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 2166 . 2  |-  ( A  e.  B  <->  E. z
( z  =  A  /\  z  e.  B
) )
2 nfcv 2312 . . . . 5  |-  F/_ x
z
3 nfnfc.1 . . . . 5  |-  F/_ x A
42, 3nfeq 2320 . . . 4  |-  F/ x  z  =  A
5 nfeq.2 . . . . 5  |-  F/_ x B
65nfcri 2306 . . . 4  |-  F/ x  z  e.  B
74, 6nfan 1558 . . 3  |-  F/ x
( z  =  A  /\  z  e.  B
)
87nfex 1630 . 2  |-  F/ x E. z ( z  =  A  /\  z  e.  B )
91, 8nfxfr 1467 1  |-  F/ x  A  e.  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348   F/wnf 1453   E.wex 1485    e. wcel 2141   F/_wnfc 2299
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301
This theorem is referenced by:  nfel1  2323  nfel2  2325  nfnel  2442  elabgf  2872  elrabf  2884  sbcel12g  3064  nfdisjv  3978  rabxfrd  4454  ffnfvf  5655  mptelixpg  6712  elabgft1  13813  elabgf2  13815  bj-rspgt  13821
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