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Theorem nfeld 2272
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 2111 . 2  |-  ( A  e.  B  <->  E. y
( y  =  A  /\  y  e.  B
) )
2 nfv 1491 . . 3  |-  F/ y
ph
3 nfcvd 2257 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfeqd.1 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeqd 2271 . . . 4  |-  ( ph  ->  F/ x  y  =  A )
6 nfeqd.2 . . . . 5  |-  ( ph  -> 
F/_ x B )
76nfcrd 2270 . . . 4  |-  ( ph  ->  F/ x  y  e.  B )
85, 7nfand 1530 . . 3  |-  ( ph  ->  F/ x ( y  =  A  /\  y  e.  B ) )
92, 8nfexd 1717 . 2  |-  ( ph  ->  F/ x E. y
( y  =  A  /\  y  e.  B
) )
101, 9nfxfrd 1434 1  |-  ( ph  ->  F/ x  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314   F/wnf 1419   E.wex 1451    e. wcel 1463   F/_wnfc 2243
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-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-cleq 2108  df-clel 2111  df-nfc 2245
This theorem is referenced by:  nfneld  2386  nfraldxy  2442  nfrexdxy  2443  nfreudxy  2579  nfsbc1d  2896  nfsbcd  2899  sbcrext  2956  nfbrd  3941  nfriotadxy  5704  nfixpxy  6577
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