Theorem nfcvb 5338

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 2406. (New usage is discouraged.)

Assertion

Ref	Expression
nfcvb	⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem nfcvb

Step	Hyp	Ref	Expression
1		nfnid 5337	. . . 4 ⊢ ¬ Ⅎ𝑦𝑦
2		eqidd 2766	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑦)
3	2	drnfc1 2946	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑦))
4	1, 3	mtbiri 330	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ Ⅎ𝑥𝑦)
5	4	con2i 140	. 2 ⊢ (Ⅎ𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
6		nfcvf 2953	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
7	5, 6	impbii 212	1 ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1561  Ⅎwnfc 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2406  ax-ext 2737  ax-nul 5261  ax-pow 5327

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-cleq 2757  df-nfc 2914

This theorem is referenced by: (None)