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Theorem nfald 1871
Description: If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
Hypotheses
Ref Expression
nfald.1  |-  F/ y
ph
nfald.2  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfald  |-  ( ph  ->  F/ x A. y ps )

Proof of Theorem nfald
StepHypRef Expression
1 nfald.1 . . 3  |-  F/ y
ph
2 nfald.2 . . 3  |-  ( ph  ->  F/ x ps )
31, 2alrimi 1781 . 2  |-  ( ph  ->  A. y F/ x ps )
4 nfnf1 1808 . . . 4  |-  F/ x F/ x ps
54nfal 1864 . . 3  |-  F/ x A. y F/ x ps
6 hba1 1804 . . . 4  |-  ( A. y F/ x ps  ->  A. y A. y F/ x ps )
7 sp 1763 . . . . 5  |-  ( A. y F/ x ps  ->  F/ x ps )
87nfrd 1779 . . . 4  |-  ( A. y F/ x ps  ->  ( ps  ->  A. x ps ) )
96, 8hbald 1755 . . 3  |-  ( A. y F/ x ps  ->  ( A. y ps  ->  A. x A. y ps ) )
105, 9nfd 1782 . 2  |-  ( A. y F/ x ps  ->  F/ x A. y ps )
113, 10syl 16 1  |-  ( ph  ->  F/ x A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   F/wnf 1553
This theorem is referenced by:  nfexd  1873  dvelimhw  1876  ax12olem4  2008  nfald2  2064  nfsb4tOLD  2128  nfeqd  2586  axrepndlem1  8467  axrepndlem2  8468  axunnd  8471  axpowndlem2  8473  axpowndlem4  8475  axregndlem2  8478  axinfndlem1  8480  axinfnd  8481  axacndlem4  8485  axacndlem5  8486  axacnd  8487
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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