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

Theorem nfnid 4927
Description: A setvar variable is not free from itself. The proof relies on dtru 4887, that is, it is not true in a one-element domain. (Contributed by Mario Carneiro, 8-Oct-2016.)
Assertion
Ref Expression
nfnid ¬ 𝑥𝑥

Proof of Theorem nfnid
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dtru 4887 . . 3 ¬ ∀𝑧 𝑧 = 𝑤
2 ax-ext 2631 . . . . 5 (∀𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
32sps 2093 . . . 4 (∀𝑤𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
43alimi 1779 . . 3 (∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤) → ∀𝑧 𝑧 = 𝑤)
51, 4mto 188 . 2 ¬ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤)
6 df-nfc 2782 . . 3 (𝑥𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
7 sbnf2 2467 . . . . 5 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥))
8 elsb4 2463 . . . . . . 7 ([𝑧 / 𝑥]𝑦𝑥𝑦𝑧)
9 elsb4 2463 . . . . . . 7 ([𝑤 / 𝑥]𝑦𝑥𝑦𝑤)
108, 9bibi12i 328 . . . . . 6 (([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ (𝑦𝑧𝑦𝑤))
11102albii 1788 . . . . 5 (∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
127, 11bitri 264 . . . 4 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
1312albii 1787 . . 3 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤))
14 alrot3 2078 . . 3 (∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤) ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
156, 13, 143bitri 286 . 2 (𝑥𝑥 ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
165, 15mtbir 312 1 ¬ 𝑥𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1521  wnf 1748  [wsb 1937  wnfc 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-nfc 2782
This theorem is referenced by:  nfcvb  4928
  Copyright terms: Public domain W3C validator