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Theorem nfexd 1873
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 1551 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
2 nfald.1 . . . 4  |-  F/ y
ph
3 nfald.2 . . . . 5  |-  ( ph  ->  F/ x ps )
43nfnd 1809 . . . 4  |-  ( ph  ->  F/ x  -.  ps )
52, 4nfald 1871 . . 3  |-  ( ph  ->  F/ x A. y  -.  ps )
65nfnd 1809 . 2  |-  ( ph  ->  F/ x  -.  A. y  -.  ps )
71, 6nfxfrd 1580 1  |-  ( ph  ->  F/ x E. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem is referenced by:  nfeld  2587  axrepndlem1  8467  axrepndlem2  8468  axunndlem1  8470  axunnd  8471  axpowndlem2  8473  axpowndlem3  8474  axpowndlem4  8475  axregndlem2  8478  axinfndlem1  8480  axinfnd  8481  axacndlem4  8485  axacndlem5  8486  axacnd  8487  19.9d2rf  23968  hbexg  28643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
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