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Theorem nfsb4t 1932
Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1930). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.)
Assertion
Ref Expression
nfsb4t (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))

Proof of Theorem nfsb4t
StepHypRef Expression
1 nfnf1 1477 . . . . 5 𝑧𝑧𝜑
21nfal 1509 . . . 4 𝑧𝑥𝑧𝜑
3 nfnae 1651 . . . 4 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
42, 3nfan 1498 . . 3 𝑧(∀𝑥𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
5 df-nf 1391 . . . . . 6 (Ⅎ𝑧𝜑 ↔ ∀𝑧(𝜑 → ∀𝑧𝜑))
65albii 1400 . . . . 5 (∀𝑥𝑧𝜑 ↔ ∀𝑥𝑧(𝜑 → ∀𝑧𝜑))
7 hbsb4t 1931 . . . . 5 (∀𝑥𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
86, 7sylbi 119 . . . 4 (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
98imp 122 . . 3 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))
104, 9nfd 1457 . 2 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
1110ex 113 1 (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1283  wnf 1390  [wsb 1686
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-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687
This theorem is referenced by:  dvelimdf  1934
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