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Theorem nfexd 1776
Description: If  x is not free in  ph, it is not free in  E. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
nfald.1  |-  F/ y
ph
nfald.2  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfexd  |-  ( ph  ->  F/ x E. y ps )

Proof of Theorem nfexd
StepHypRef Expression
1 df-ex 1529 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
2 nfald.1 . . . 4  |-  F/ y
ph
3 nfald.2 . . . . 5  |-  ( ph  ->  F/ x ps )
43nfnd 1760 . . . 4  |-  ( ph  ->  F/ x  -.  ps )
52, 4nfald 1775 . . 3  |-  ( ph  ->  F/ x A. y  -.  ps )
65nfnd 1760 . 2  |-  ( ph  ->  F/ x  -.  A. y  -.  ps )
71, 6nfxfrd 1558 1  |-  ( ph  ->  F/ x E. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527   E.wex 1528   F/wnf 1531
This theorem is referenced by:  nfeld  2434  axrepndlem1  8214  axrepndlem2  8215  axunndlem1  8217  axunnd  8218  axpowndlem2  8220  axpowndlem3  8221  axpowndlem4  8222  axregndlem2  8225  axinfndlem1  8227  axinfnd  8228  axacndlem4  8232  axacndlem5  8233  axacnd  8234  hbexg  27695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-ex 1529  df-nf 1532
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