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Theorem nfeld 2193
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 2036 . 2  |-  ( A  e.  B  <->  E. y
( y  =  A  /\  y  e.  B
) )
2 nfv 1421 . . 3  |-  F/ y
ph
3 nfcvd 2179 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfeqd.1 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeqd 2192 . . . 4  |-  ( ph  ->  F/ x  y  =  A )
6 nfeqd.2 . . . . 5  |-  ( ph  -> 
F/_ x B )
76nfcrd 2191 . . . 4  |-  ( ph  ->  F/ x  y  e.  B )
85, 7nfand 1460 . . 3  |-  ( ph  ->  F/ x ( y  =  A  /\  y  e.  B ) )
92, 8nfexd 1644 . 2  |-  ( ph  ->  F/ x E. y
( y  =  A  /\  y  e.  B
) )
101, 9nfxfrd 1364 1  |-  ( ph  ->  F/ x  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   F/wnf 1349   E.wex 1381    e. wcel 1393   F/_wnfc 2165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167
This theorem is referenced by:  nfneld  2305  nfraldxy  2356  nfrexdxy  2357  nfreudxy  2483  nfsbc1d  2780  nfsbcd  2783  sbcrext  2835  nfbrd  3807  nfriotadxy  5476
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