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	⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem nfcvb

Step	Hyp	Ref	Expression
1		nfnid 5247	. . . 4 ⊢ ¬ Ⅎ𝑦𝑦
2		eqidd 2759	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑦)
3	2	drnfc1 2938	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑦))
4	1, 3	mtbiri 330	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ Ⅎ𝑥𝑦)
5	4	con2i 141	. 2 ⊢ (Ⅎ𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
6		nfcvf 2945	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
7	5, 6	impbii 212	1 ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

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)