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Theorem nfald 1690
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 . . . 4  |-  F/ y
ph
21nfri 1457 . . 3  |-  ( ph  ->  A. y ph )
3 nfald.2 . . 3  |-  ( ph  ->  F/ x ps )
42, 3alrimih 1403 . 2  |-  ( ph  ->  A. y F/ x ps )
5 nfnf1 1481 . . . 4  |-  F/ x F/ x ps
65nfal 1513 . . 3  |-  F/ x A. y F/ x ps
7 hba1 1478 . . . 4  |-  ( A. y F/ x ps  ->  A. y A. y F/ x ps )
8 sp 1446 . . . . 5  |-  ( A. y F/ x ps  ->  F/ x ps )
98nfrd 1458 . . . 4  |-  ( A. y F/ x ps  ->  ( ps  ->  A. x ps ) )
107, 9hbald 1425 . . 3  |-  ( A. y F/ x ps  ->  ( A. y ps  ->  A. x A. y ps ) )
116, 10nfd 1461 . 2  |-  ( A. y F/ x ps  ->  F/ x A. y ps )
124, 11syl 14 1  |-  ( ph  ->  F/ x A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287   F/wnf 1394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  dvelimALT  1934  dvelimfv  1935  nfeudv  1963  nfeqd  2243  nfraldxy  2410  nfiotadxy  4970  bdsepnft  11435  strcollnft  11536
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