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Theorem nfald 1640
 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 1409 . . 3 (φyφ)
3 nfald.2 . . 3 (φ → Ⅎxψ)
42, 3alrimih 1355 . 2 (φyxψ)
5 nfnf1 1433 . . . 4 xxψ
65nfal 1465 . . 3 xyxψ
7 hba1 1430 . . . 4 (yxψyyxψ)
8 sp 1398 . . . . 5 (yxψ → Ⅎxψ)
98nfrd 1410 . . . 4 (yxψ → (ψxψ))
107, 9hbald 1377 . . 3 (yxψ → (yψxyψ))
116, 10nfd 1413 . 2 (yxψ → Ⅎxyψ)
124, 11syl 14 1 (φ → Ⅎxyψ)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240  Ⅎwnf 1346 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 1333  ax-7 1334  ax-gen 1335  ax-4 1397  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-nf 1347 This theorem is referenced by:  dvelimALT  1883  dvelimfv  1884  nfeudv  1912  nfeqd  2189  nfraldxy  2350  nfiotadxy  4812  bdsepnft  8940  strcollnft  9033
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