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Theorem nfcvb 5026
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 5025 . . . 4 ¬ 𝑦𝑦
2 eqidd 2771 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑦)
32drnfc1 2930 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
41, 3mtbiri 316 . . 3 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
54con2i 136 . 2 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
6 nfcvf 2936 . 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 1628  wnfc 2899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920  ax-pow 4971
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-cleq 2763  df-clel 2766  df-nfc 2901
This theorem is referenced by: (None)
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