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Theorem nfcvb 4819
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 4818 . . . 4 ¬ 𝑦𝑦
2 eqidd 2610 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑦)
32drnfc1 2767 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
41, 3mtbiri 315 . . 3 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
54con2i 132 . 2 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
6 nfcvf 2773 . 2 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
75, 6impbii 197 1 (𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wal 1472  wnfc 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712  ax-pow 4764
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-cleq 2602  df-clel 2605  df-nfc 2739
This theorem is referenced by: (None)
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