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Theorem nfsb4t 2532
Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (closed form of nfsb4 2533). (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 2243 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
21sps 2174 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
32drnf2 2458 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
43biimpd 230 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
54spsd 2176 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
65impcom 408 . . 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 2149 . . . . 5 𝑧𝑧𝜑
98nfal 2333 . . . 4 𝑧𝑥𝑧𝜑
10 nfnae 2448 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
119, 10nfan 1891 . . 3 𝑧(∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
12 nfa1 2146 . . . 4 𝑥𝑥𝑧𝜑
13 nfnae 2448 . . . 4 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
1412, 13nfan 1891 . . 3 𝑥(∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
15 sp 2172 . . . 4 (∀𝑥𝑧𝜑 → Ⅎ𝑧𝜑)
1615adantr 481 . . 3 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜑)
17 nfsb2 2515 . . . 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 2463 . 2 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
217, 20pm2.61dan 809 1 (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526  wnf 1775  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by:  nfsb4  2533  nfsbd  2557  ichnfimlem1  43498
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