Theorem nfcvb 5337

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 2401. (New usage is discouraged.)

Assertion

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

Proof of Theorem nfcvb

Step	Hyp	Ref	Expression
1		nfnid 5336	. . . 4 ⊢ ¬ Ⅎ𝑦𝑦
2		eqidd 2761	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑦)
3	2	drnfc1 2941	. . . 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 2948	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
7	5, 6	impbii 212	1 ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1567  Ⅎwnfc 2907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732  ax-nul 5259  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-cleq 2752  df-nfc 2909

This theorem is used by: (None)