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 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. (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	⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem nfcvb

Step	Hyp	Ref	Expression
1		nfnid 5369	. . . 4 ⊢ ¬ Ⅎ𝑦𝑦
2		eqidd 2728	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑦)
3	2	drnfc1 2917	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑦))
4	1, 3	mtbiri 327	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ Ⅎ𝑥𝑦)
5	4	con2i 139	. 2 ⊢ (Ⅎ𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
6		nfcvf 2927	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
7	5, 6	impbii 208	1 ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1532  Ⅎwnfc 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)