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Theorem nfeld 2209
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 2052 . 2  |-  ( A  e.  B  <->  E. y
( y  =  A  /\  y  e.  B
) )
2 nfv 1437 . . 3  |-  F/ y
ph
3 nfcvd 2195 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfeqd.1 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeqd 2208 . . . 4  |-  ( ph  ->  F/ x  y  =  A )
6 nfeqd.2 . . . . 5  |-  ( ph  -> 
F/_ x B )
76nfcrd 2207 . . . 4  |-  ( ph  ->  F/ x  y  e.  B )
85, 7nfand 1476 . . 3  |-  ( ph  ->  F/ x ( y  =  A  /\  y  e.  B ) )
92, 8nfexd 1660 . 2  |-  ( ph  ->  F/ x E. y
( y  =  A  /\  y  e.  B
) )
101, 9nfxfrd 1380 1  |-  ( ph  ->  F/ x  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ 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-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052  df-nfc 2183
This theorem is referenced by:  nfneld  2322  nfraldxy  2373  nfrexdxy  2374  nfreudxy  2500  nfsbc1d  2802  nfsbcd  2805  sbcrext  2862  nfbrd  3834  nfriotadxy  5503
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