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Theorem nfimd 1518
Description: If in a context  x is not free in  ps and  ch, it is not free in  ( ps  ->  ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
Hypotheses
Ref Expression
nfimd.1  |-  ( ph  ->  F/ x ps )
nfimd.2  |-  ( ph  ->  F/ x ch )
Assertion
Ref Expression
nfimd  |-  ( ph  ->  F/ x ( ps 
->  ch ) )

Proof of Theorem nfimd
StepHypRef Expression
1 nfimd.1 . 2  |-  ( ph  ->  F/ x ps )
2 nfimd.2 . 2  |-  ( ph  ->  F/ x ch )
3 nfnf1 1477 . . . . 5  |-  F/ x F/ x ps
43nfri 1453 . . . 4  |-  ( F/ x ps  ->  A. x F/ x ps )
5 nfnf1 1477 . . . . 5  |-  F/ x F/ x ch
65nfri 1453 . . . 4  |-  ( F/ x ch  ->  A. x F/ x ch )
7 nfr 1452 . . . . . 6  |-  ( F/ x ch  ->  ( ch  ->  A. x ch )
)
87imim2d 53 . . . . 5  |-  ( F/ x ch  ->  (
( ps  ->  ch )  ->  ( ps  ->  A. x ch ) ) )
9 19.21t 1515 . . . . . 6  |-  ( F/ x ps  ->  ( A. x ( ps  ->  ch )  <->  ( ps  ->  A. x ch ) ) )
109biimprd 156 . . . . 5  |-  ( F/ x ps  ->  (
( ps  ->  A. x ch )  ->  A. x
( ps  ->  ch ) ) )
118, 10syl9r 72 . . . 4  |-  ( F/ x ps  ->  ( F/ x ch  ->  (
( ps  ->  ch )  ->  A. x ( ps 
->  ch ) ) ) )
124, 6, 11alrimdh 1409 . . 3  |-  ( F/ x ps  ->  ( F/ x ch  ->  A. x
( ( ps  ->  ch )  ->  A. x
( ps  ->  ch ) ) ) )
13 df-nf 1391 . . 3  |-  ( F/ x ( ps  ->  ch )  <->  A. x ( ( ps  ->  ch )  ->  A. x ( ps 
->  ch ) ) )
1412, 13syl6ibr 160 . 2  |-  ( F/ x ps  ->  ( F/ x ch  ->  F/ x ( ps  ->  ch ) ) )
151, 2, 14sylc 61 1  |-  ( ph  ->  F/ x ( ps 
->  ch ) )
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-gen 1379  ax-4 1441  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  nfbid  1521  dvelimALT  1929  dvelimfv  1930  dvelimor  1937  nfmod  1960  nfraldxy  2404  cbvrald  11031
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