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Theorem nfald 1625
Description: If x is not free in φ, it is not free in yφ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
Hypotheses
Ref Expression
nfald.1 yφ
nfald.2 (φ → Ⅎxψ)
Assertion
Ref Expression
nfald (φ → Ⅎxyψ)

Proof of Theorem nfald
StepHypRef Expression
1 nfald.1 . . . 4 yφ
21nfri 1393 . . 3 (φyφ)
3 nfald.2 . . 3 (φ → Ⅎxψ)
42, 3alrimih 1338 . 2 (φyxψ)
5 nfnf1 1418 . . . 4 xxψ
65nfal 1450 . . 3 xyxψ
7 hba1 1415 . . . 4 (yxψyyxψ)
8 sp 1382 . . . . 5 (yxψ → Ⅎxψ)
98nfrd 1394 . . . 4 (yxψ → (ψxψ))
107, 9hbald 1361 . . 3 (yxψ → (yψxyψ))
116, 10nfd 1397 . 2 (yxψ → Ⅎxyψ)
124, 11syl 14 1 (φ → Ⅎxyψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226  wnf 1329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-4 1381  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-nf 1330
This theorem is referenced by:  dvelimALT  1868  dvelimfv  1869  nfeudv  1897  nfeqd  2174  nfraldxy  2334  nfiotadxy  4797  bdsepnft  7113  strcollnft  7202
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