MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb4b Structured version   Visualization version   GIF version

Theorem sb4b 2474
Description: Simplified definition of substitution when variables are distinct. Version of sb6 2089 with a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 27-May-1997.) Revise df-sb 2069. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
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 nfna1 2150 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑡
2 nfeqf2 2376 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡)
31, 2nfan1 2194 . . . . 5 𝑥(¬ ∀𝑥 𝑥 = 𝑡𝑦 = 𝑡)
4 equequ2 2030 . . . . . . 7 (𝑦 = 𝑡 → (𝑥 = 𝑦𝑥 = 𝑡))
54imbi1d 342 . . . . . 6 (𝑦 = 𝑡 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
65adantl 483 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑡𝑦 = 𝑡) → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
73, 6albid 2216 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑡𝑦 = 𝑡) → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
87pm5.74da 803 . . 3 (¬ ∀𝑥 𝑥 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
98albidv 1924 . 2 (¬ ∀𝑥 𝑥 = 𝑡 → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
10 df-sb 2069 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
11 ax6ev 1974 . . . 4 𝑦 𝑦 = 𝑡
1211a1bi 363 . . 3 (∀𝑥(𝑥 = 𝑡𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
13 19.23v 1946 . . 3 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
1412, 13bitr4i 278 . 2 (∀𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
159, 10, 143bitr4g 314 1 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540  wex 1782  [wsb 2068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-12 2172  ax-13 2371
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069
This theorem is referenced by:  sb3b  2475  sb2  2478  sb4a  2479  hbsb2  2481  dfsb2  2492  sbcom3  2505  sbal1  2528  sbal2  2529  wl-2sb6d  36059  wl-sbalnae  36063
  Copyright terms: Public domain W3C validator