Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsb4t Structured version   Visualization version   GIF version

Theorem nfsb4t 2377
 Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2378). (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 2097 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
21sps 2043 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
32drnf2 2318 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
43biimpd 218 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
54spsd 2045 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
65impcom 445 . . 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 2018 . . . . 5 𝑧𝑧𝜑
98nfal 2139 . . . 4 𝑧𝑥𝑧𝜑
10 nfnae 2306 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
119, 10nfan 1816 . . 3 𝑧(∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
12 nfa1 2015 . . . 4 𝑥𝑥𝑧𝜑
13 nfnae 2306 . . . 4 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
1412, 13nfan 1816 . . 3 𝑥(∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
15 sp 2041 . . . 4 (∀𝑥𝑧𝜑 → Ⅎ𝑧𝜑)
1615adantr 480 . . 3 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜑)
17 nfsb2 2348 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
1817adantl 481 . . 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 2323 . 2 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
217, 20pm2.61dan 828 1 (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  Ⅎwnf 1699  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  nfsb4  2378  nfsbd  2430
 Copyright terms: Public domain W3C validator