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Theorem nfcvb 5248
 Description: The "distinctor" expression ¬ ∀𝑥𝑥 = 𝑦, stating that 𝑥 and 𝑦 are not the same variable, can be written in terms of Ⅎ in the obvious way. This theorem is not true in a one-element domain, because then Ⅎ𝑥𝑦 and ∀𝑥𝑥 = 𝑦 will both be true. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
Assertion
Ref Expression
nfcvb (𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem nfcvb
StepHypRef Expression
1 nfnid 5247 . . . 4 ¬ 𝑦𝑦
2 eqidd 2759 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑦)
32drnfc1 2938 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
41, 3mtbiri 330 . . 3 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
54con2i 141 . 2 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
6 nfcvf 2945 . 2 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
75, 6impbii 212 1 (𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1536  Ⅎwnfc 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729  ax-nul 5179  ax-pow 5237 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-cleq 2750  df-clel 2830  df-nfc 2901 This theorem is referenced by: (None)
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