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Theorem nfnf 2007
Description: If  x is not free in  ph, it is not free in  F/ y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
Hypothesis
Ref Expression
nfal.1  |-  F/ x ph
Assertion
Ref Expression
nfnf  |-  F/ x F/ y ph

Proof of Theorem nfnf
StepHypRef Expression
1 df-nf 1664 . 2  |-  ( F/ y ph  <->  A. y
( ph  ->  A. y ph ) )
2 nfal.1 . . . 4  |-  F/ x ph
32nfal 2005 . . . 4  |-  F/ x A. y ph
42, 3nfim 1978 . . 3  |-  F/ x
( ph  ->  A. y ph )
54nfal 2005 . 2  |-  F/ x A. y ( ph  ->  A. y ph )
61, 5nfxfr 1692 1  |-  F/ x F/ y ph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435   F/wnf 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907
This theorem depends on definitions:  df-bi 188  df-ex 1660  df-nf 1664
This theorem is referenced by:  nfnfc  2600  nfnfcALT  2601  bj-nfnfc  31213  bj-nfcf  31277
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