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Theorem sb4b 2457
 Description: Simplified definition of substitution when variables are distinct. Version of sb6 2066 with a distinctor. (Contributed by NM, 27-May-1997.) Revise df-sb 2043. (Revised by Wolf Lammen, 25-Jul-2023.)
Assertion
Ref Expression
sb4b (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))

Proof of Theorem sb4b
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfeqf2 2349 . . . 4 (¬ ∀𝑥 𝑥 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡)
2 nfnf1 2124 . . . . . . 7 𝑥𝑥 𝑦 = 𝑡
3 id 22 . . . . . . 7 (Ⅎ𝑥 𝑦 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡)
42, 3nfan1 2165 . . . . . 6 𝑥(Ⅎ𝑥 𝑦 = 𝑡𝑦 = 𝑡)
5 equequ2 2010 . . . . . . . 8 (𝑦 = 𝑡 → (𝑥 = 𝑦𝑥 = 𝑡))
65imbi1d 343 . . . . . . 7 (𝑦 = 𝑡 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
76adantl 482 . . . . . 6 ((Ⅎ𝑥 𝑦 = 𝑡𝑦 = 𝑡) → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
84, 7albid 2189 . . . . 5 ((Ⅎ𝑥 𝑦 = 𝑡𝑦 = 𝑡) → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
98pm5.74da 800 . . . 4 (Ⅎ𝑥 𝑦 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
101, 9syl 17 . . 3 (¬ ∀𝑥 𝑥 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
1110albidv 1898 . 2 (¬ ∀𝑥 𝑥 = 𝑡 → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
12 df-sb 2043 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
13 ax6ev 1949 . . . 4 𝑦 𝑦 = 𝑡
1413a1bi 364 . . 3 (∀𝑥(𝑥 = 𝑡𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
15 19.23v 1920 . . 3 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
1614, 15bitr4i 279 . 2 (∀𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
1711, 12, 163bitr4g 315 1 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520  ∃wex 1761  Ⅎwnf 1765  [wsb 2042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043 This theorem is referenced by:  sb2  2458  sb4OLD  2460  sb1  2461  sb4a  2463  hbsb2  2475  dfsb2  2486  sbcom3  2502  sbal1  2525  sbal2  2526  sbal2OLD  2527  sbal2OLDOLD  2528  wl-2sb6d  34344  wl-sbalnae  34348
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