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Theorem nfcvb 5370
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.) Usage of this theorem is discouraged because it depends on ax-13 2366. (New usage is discouraged.)
Assertion
Ref Expression
nfcvb (𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem nfcvb
StepHypRef Expression
1 nfnid 5369 . . . 4 ¬ 𝑦𝑦
2 eqidd 2728 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑦)
32drnfc1 2917 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
41, 3mtbiri 327 . . 3 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
54con2i 139 . 2 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
6 nfcvf 2927 . 2 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
75, 6impbii 208 1 (𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1532  wnfc 2878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-13 2366  ax-ext 2698  ax-nul 5300  ax-pow 5359
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-cleq 2719  df-nfc 2880
This theorem is referenced by: (None)
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