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Theorem nfsb4t 2387
Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2388). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
Assertion
Ref Expression
nfsb4t (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))

Proof of Theorem nfsb4t
StepHypRef Expression
1 sbequ12 2109 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
21sps 2053 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
32drnf2 2328 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
43biimpd 219 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
54spsd 2055 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
65impcom 446 . . 3 ((∀𝑥𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
76a1d 25 . 2 ((∀𝑥𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
8 nfnf1 2029 . . . . 5 𝑧𝑧𝜑
98nfal 2151 . . . 4 𝑧𝑥𝑧𝜑
10 nfnae 2316 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
119, 10nfan 1826 . . 3 𝑧(∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
12 nfa1 2026 . . . 4 𝑥𝑥𝑧𝜑
13 nfnae 2316 . . . 4 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
1412, 13nfan 1826 . . 3 𝑥(∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
15 sp 2051 . . . 4 (∀𝑥𝑧𝜑 → Ⅎ𝑧𝜑)
1615adantr 481 . . 3 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜑)
17 nfsb2 2358 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
1817adantl 482 . . 3 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
191a1i 11 . . 3 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)))
2011, 14, 16, 18, 19dvelimdf 2333 . 2 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
217, 20pm2.61dan 831 1 (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479  wnf 1706  [wsb 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879
This theorem is referenced by:  nfsb4  2388  nfsbd  2440
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