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Theorem nfnid 4823
 Description: A setvar variable is not free from itself. The proof relies on dtru 4783, 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 4783 . . 3 ¬ ∀𝑧 𝑧 = 𝑤
2 ax-ext 2590 . . . . 5 (∀𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
32sps 2043 . . . 4 (∀𝑤𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
43alimi 1730 . . 3 (∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤) → ∀𝑧 𝑧 = 𝑤)
51, 4mto 187 . 2 ¬ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤)
6 df-nfc 2740 . . 3 (𝑥𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
7 sbnf2 2427 . . . . 5 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥))
8 elsb4 2423 . . . . . . 7 ([𝑧 / 𝑥]𝑦𝑥𝑦𝑧)
9 elsb4 2423 . . . . . . 7 ([𝑤 / 𝑥]𝑦𝑥𝑦𝑤)
108, 9bibi12i 328 . . . . . 6 (([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ (𝑦𝑧𝑦𝑤))
11102albii 1738 . . . . 5 (∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
127, 11bitri 263 . . . 4 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
1312albii 1737 . . 3 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤))
14 alrot3 2025 . . 3 (∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤) ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
156, 13, 143bitri 285 . 2 (𝑥𝑥 ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
165, 15mtbir 312 1 ¬ 𝑥𝑥
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699  [wsb 1867  Ⅎwnfc 2738 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717  ax-pow 4769 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  df-nfc 2740 This theorem is referenced by:  nfcvb  4824
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