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Theorem nfald 1684
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 1453 . . 3 (𝜑 → ∀𝑦𝜑)
3 nfald.2 . . 3 (𝜑 → Ⅎ𝑥𝜓)
42, 3alrimih 1399 . 2 (𝜑 → ∀𝑦𝑥𝜓)
5 nfnf1 1477 . . . 4 𝑥𝑥𝜓
65nfal 1509 . . 3 𝑥𝑦𝑥𝜓
7 hba1 1474 . . . 4 (∀𝑦𝑥𝜓 → ∀𝑦𝑦𝑥𝜓)
8 sp 1442 . . . . 5 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝜓)
98nfrd 1454 . . . 4 (∀𝑦𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
107, 9hbald 1421 . . 3 (∀𝑦𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))
116, 10nfd 1457 . 2 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝑦𝜓)
124, 11syl 14 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  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  4894  bdsepnft  10821  strcollnft  10922
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