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Theorem nfnid 5249
 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 5244 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 5244 . . 3 ¬ ∀𝑧 𝑧 = 𝑤
2 ax-ext 2730 . . . . 5 (∀𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
32sps 2183 . . . 4 (∀𝑤𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
43alimi 1814 . . 3 (∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤) → ∀𝑧 𝑧 = 𝑤)
51, 4mto 200 . 2 ¬ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤)
6 df-nfc 2902 . . 3 (𝑥𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
7 sbnf2 2367 . . . . 5 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥))
8 elsb4 2128 . . . . . . 7 ([𝑧 / 𝑥]𝑦𝑥𝑦𝑧)
9 elsb4 2128 . . . . . . 7 ([𝑤 / 𝑥]𝑦𝑥𝑦𝑤)
108, 9bibi12i 343 . . . . . 6 (([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ (𝑦𝑧𝑦𝑤))
11102albii 1823 . . . . 5 (∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
127, 11bitri 278 . . . 4 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
1312albii 1822 . . 3 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤))
14 alrot3 2162 . . 3 (∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤) ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
156, 13, 143bitri 300 . 2 (𝑥𝑥 ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
165, 15mtbir 326 1 ¬ 𝑥𝑥
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1537  Ⅎwnf 1786  [wsb 2070  Ⅎwnfc 2900 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-nul 5181  ax-pow 5239 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-nfc 2902 This theorem is referenced by:  nfcvb  5250
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