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Theorem nfcvb 4896
Description: The "distinctor" expression ¬ ∀𝑥𝑥 = 𝑦, stating that 𝑥 and 𝑦 are not the same variable, can be written in terms of in the obvious way. This theorem is not true in a one-element domain, because then 𝑥𝑦 and 𝑥𝑥 = 𝑦 will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.)
Assertion
Ref Expression
nfcvb (𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem nfcvb
StepHypRef Expression
1 nfnid 4895 . . . 4 ¬ 𝑦𝑦
2 eqidd 2622 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑦)
32drnfc1 2781 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
41, 3mtbiri 317 . . 3 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
54con2i 134 . 2 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
6 nfcvf 2787 . 2 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
75, 6impbii 199 1 (𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1480  wnfc 2750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787  ax-pow 4841
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-cleq 2614  df-clel 2617  df-nfc 2752
This theorem is referenced by: (None)
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