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Theorem nfel 2202
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 2052 . 2  |-  ( A  e.  B  <->  E. z
( z  =  A  /\  z  e.  B
) )
2 nfcv 2194 . . . . 5  |-  F/_ x
z
3 nfnfc.1 . . . . 5  |-  F/_ x A
42, 3nfeq 2201 . . . 4  |-  F/ x  z  =  A
5 nfeq.2 . . . . 5  |-  F/_ x B
65nfcri 2188 . . . 4  |-  F/ x  z  e.  B
74, 6nfan 1473 . . 3  |-  F/ x
( z  =  A  /\  z  e.  B
)
87nfex 1544 . 2  |-  F/ x E. z ( z  =  A  /\  z  e.  B )
91, 8nfxfr 1379 1  |-  F/ x  A  e.  B
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259   F/wnf 1365   E.wex 1397    e. wcel 1409   F/_wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183
This theorem is referenced by:  nfel1  2204  nfel2  2206  nfnel  2321  elabgf  2708  elrabf  2719  sbcel12g  2893  nfdisjv  3785  rabxfrd  4229  ffnfvf  5352  elabgft1  10304  elabgf2  10306  bj-rspgt  10312
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