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Theorem nfnid 5267
Description: A setvar variable is not free from itself. This theorem is not true in a one-element domain, as illustrated by the use of dtru 5262 in its proof. (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 5262 . . 3 ¬ ∀𝑧 𝑧 = 𝑤
2 ax-ext 2790 . . . . 5 (∀𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
32sps 2174 . . . 4 (∀𝑤𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
43alimi 1803 . . 3 (∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤) → ∀𝑧 𝑧 = 𝑤)
51, 4mto 198 . 2 ¬ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤)
6 df-nfc 2960 . . 3 (𝑥𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
7 sbnf2 2368 . . . . 5 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥))
8 elsb4 2121 . . . . . . 7 ([𝑧 / 𝑥]𝑦𝑥𝑦𝑧)
9 elsb4 2121 . . . . . . 7 ([𝑤 / 𝑥]𝑦𝑥𝑦𝑤)
108, 9bibi12i 341 . . . . . 6 (([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ (𝑦𝑧𝑦𝑤))
11102albii 1812 . . . . 5 (∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
127, 11bitri 276 . . . 4 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
1312albii 1811 . . 3 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤))
14 alrot3 2154 . . 3 (∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤) ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
156, 13, 143bitri 298 . 2 (𝑥𝑥 ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
165, 15mtbir 324 1 ¬ 𝑥𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wal 1526  wnf 1775  [wsb 2060  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201  ax-pow 5257
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-nfc 2960
This theorem is referenced by:  nfcvb  5268
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