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Theorem nfeld 2244
Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfeqd.1  |-  ( ph  -> 
F/_ x A )
nfeqd.2  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfeld  |-  ( ph  ->  F/ x  A  e.  B )

Proof of Theorem nfeld
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-clel 2084 . 2  |-  ( A  e.  B  <->  E. y
( y  =  A  /\  y  e.  B
) )
2 nfv 1466 . . 3  |-  F/ y
ph
3 nfcvd 2229 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfeqd.1 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeqd 2243 . . . 4  |-  ( ph  ->  F/ x  y  =  A )
6 nfeqd.2 . . . . 5  |-  ( ph  -> 
F/_ x B )
76nfcrd 2242 . . . 4  |-  ( ph  ->  F/ x  y  e.  B )
85, 7nfand 1505 . . 3  |-  ( ph  ->  F/ x ( y  =  A  /\  y  e.  B ) )
92, 8nfexd 1691 . 2  |-  ( ph  ->  F/ x E. y
( y  =  A  /\  y  e.  B
) )
101, 9nfxfrd 1409 1  |-  ( ph  ->  F/ x  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ 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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-clel 2084  df-nfc 2217
This theorem is referenced by:  nfneld  2358  nfraldxy  2410  nfrexdxy  2411  nfreudxy  2540  nfsbc1d  2854  nfsbcd  2857  sbcrext  2914  nfbrd  3880  nfriotadxy  5598
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