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Theorem nfald 1684
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 1453 . . 3  |-  ( ph  ->  A. y ph )
3 nfald.2 . . 3  |-  ( ph  ->  F/ x ps )
42, 3alrimih 1399 . 2  |-  ( ph  ->  A. y F/ x ps )
5 nfnf1 1477 . . . 4  |-  F/ x F/ x ps
65nfal 1509 . . 3  |-  F/ x A. y F/ x ps
7 hba1 1474 . . . 4  |-  ( A. y F/ x ps  ->  A. y A. y F/ x ps )
8 sp 1442 . . . . 5  |-  ( A. y F/ x ps  ->  F/ x ps )
98nfrd 1454 . . . 4  |-  ( A. y F/ x ps  ->  ( ps  ->  A. x ps ) )
107, 9hbald 1421 . . 3  |-  ( A. y F/ x ps  ->  ( A. y ps  ->  A. x A. y ps ) )
116, 10nfd 1457 . 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 1283   F/wnf 1390
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 1377  ax-7 1378  ax-gen 1379  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  dvelimALT  1928  dvelimfv  1929  nfeudv  1957  nfeqd  2234  nfraldxy  2399  nfiotadxy  4900  bdsepnft  10836  strcollnft  10937
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