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Theorem nfeld 2315
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 2153 . 2  |-  ( A  e.  B  <->  E. y
( y  =  A  /\  y  e.  B
) )
2 nfv 1508 . . 3  |-  F/ y
ph
3 nfcvd 2300 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfeqd.1 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeqd 2314 . . . 4  |-  ( ph  ->  F/ x  y  =  A )
6 nfeqd.2 . . . . 5  |-  ( ph  -> 
F/_ x B )
76nfcrd 2313 . . . 4  |-  ( ph  ->  F/ x  y  e.  B )
85, 7nfand 1548 . . 3  |-  ( ph  ->  F/ x ( y  =  A  /\  y  e.  B ) )
92, 8nfexd 1741 . 2  |-  ( ph  ->  F/ x E. y
( y  =  A  /\  y  e.  B
) )
101, 9nfxfrd 1455 1  |-  ( ph  ->  F/ x  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335   F/wnf 1440   E.wex 1472    e. wcel 2128   F/_wnfc 2286
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-cleq 2150  df-clel 2153  df-nfc 2288
This theorem is referenced by:  nfneld  2430  nfraldw  2489  nfraldxy  2490  nfrexdxy  2491  nfreudxy  2630  nfsbc1d  2953  nfsbcd  2956  sbcrext  3014  nfsbcdw  3065  nfbrd  4009  nfriotadxy  5788  nfixpxy  6662
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