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Theorem nfald 1643
 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 1412 . . 3 (𝜑 → ∀𝑦𝜑)
3 nfald.2 . . 3 (𝜑 → Ⅎ𝑥𝜓)
42, 3alrimih 1358 . 2 (𝜑 → ∀𝑦𝑥𝜓)
5 nfnf1 1436 . . . 4 𝑥𝑥𝜓
65nfal 1468 . . 3 𝑥𝑦𝑥𝜓
7 hba1 1433 . . . 4 (∀𝑦𝑥𝜓 → ∀𝑦𝑦𝑥𝜓)
8 sp 1401 . . . . 5 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝜓)
98nfrd 1413 . . . 4 (∀𝑦𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
107, 9hbald 1380 . . 3 (∀𝑦𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))
116, 10nfd 1416 . 2 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝑦𝜓)
124, 11syl 14 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349 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 1336  ax-7 1337  ax-gen 1338  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  dvelimALT  1886  dvelimfv  1887  nfeudv  1915  nfeqd  2192  nfraldxy  2356  nfiotadxy  4870  bdsepnft  10007  strcollnft  10109
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