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Theorem nfeld 2637
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 2462 . 2  |-  ( A  e.  B  <->  E. y
( y  =  A  /\  y  e.  B
) )
2 nfv 1683 . . 3  |-  F/ y
ph
3 nfcvd 2630 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfeqd.1 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeqd 2636 . . . 4  |-  ( ph  ->  F/ x  y  =  A )
6 nfeqd.2 . . . . 5  |-  ( ph  -> 
F/_ x B )
76nfcrd 2635 . . . 4  |-  ( ph  ->  F/ x  y  e.  B )
85, 7nfand 1872 . . 3  |-  ( ph  ->  F/ x ( y  =  A  /\  y  e.  B ) )
92, 8nfexd 1899 . 2  |-  ( ph  ->  F/ x E. y
( y  =  A  /\  y  e.  B
) )
101, 9nfxfrd 1626 1  |-  ( ph  ->  F/ x  A  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596   F/wnf 1599    e. wcel 1767   F/_wnfc 2615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-cleq 2459  df-clel 2462  df-nfc 2617
This theorem is referenced by:  nfel  2642  nfneld  2811  nfrald  2849  ralcom2  3026  nfreud  3034  nfrmod  3035  nfrmo  3037  nfsbc1d  3349  nfsbcd  3352  sbcrextOLD  3413  sbcrext  3414  nfdisj  4429  nfbrd  4490  nfriotad  6251  nfixp  7485  axrepndlem2  8964  axrepnd  8965  axunnd  8967  axpowndlem2  8969  axpowndlem2OLD  8970  axpowndlem3  8971  axpowndlem3OLD  8972  axpowndlem4  8973  axpownd  8974  axregndlem2  8976  axinfndlem1  8979  axinfnd  8980  axacndlem4  8984  axacndlem5  8985  axacnd  8986
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