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

Theorem nfnid 5381
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 dtruALT2 5376 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 dtruALT2 5376 . . 3 ¬ ∀𝑧 𝑧 = 𝑤
2 ax-ext 2706 . . . . 5 (∀𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
32sps 2183 . . . 4 (∀𝑤𝑦(𝑦𝑧𝑦𝑤) → 𝑧 = 𝑤)
43alimi 1808 . . 3 (∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤) → ∀𝑧 𝑧 = 𝑤)
51, 4mto 197 . 2 ¬ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤)
6 df-nfc 2890 . . 3 (𝑥𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
7 sbnf2 2359 . . . . 5 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥))
8 elsb2 2123 . . . . . . 7 ([𝑧 / 𝑥]𝑦𝑥𝑦𝑧)
9 elsb2 2123 . . . . . . 7 ([𝑤 / 𝑥]𝑦𝑥𝑦𝑤)
108, 9bibi12i 339 . . . . . 6 (([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ (𝑦𝑧𝑦𝑤))
11102albii 1817 . . . . 5 (∀𝑧𝑤([𝑧 / 𝑥]𝑦𝑥 ↔ [𝑤 / 𝑥]𝑦𝑥) ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
127, 11bitri 275 . . . 4 (Ⅎ𝑥 𝑦𝑥 ↔ ∀𝑧𝑤(𝑦𝑧𝑦𝑤))
1312albii 1816 . . 3 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤))
14 alrot3 2158 . . 3 (∀𝑦𝑧𝑤(𝑦𝑧𝑦𝑤) ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
156, 13, 143bitri 297 . 2 (𝑥𝑥 ↔ ∀𝑧𝑤𝑦(𝑦𝑧𝑦𝑤))
165, 15mtbir 323 1 ¬ 𝑥𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wal 1535  wnf 1780  [wsb 2062  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pow 5371
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-nfc 2890
This theorem is referenced by:  nfcvb  5382
  Copyright terms: Public domain W3C validator