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Metamath Proof Explorer

Theorem nfcvb

Description: The "distinctor" expression -. A. x x = y , stating that x and y are not the same variable, can be written in terms of F/ in the obvious way. This theorem is not true in a one-element domain, because then F/_ x y and A. x x = y will both be true. (Contributed by Mario Carneiro, 8-Oct-2016) Usage of this theorem is discouraged because it depends on ax-13 . (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfnid	⊢ ¬ Ⅎ 𝑦 𝑦
2		eqidd	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝑦 = 𝑦 )
3	2	drnfc1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑥 𝑦 ↔ Ⅎ 𝑦 𝑦 ) )
4	1 3	mtbiri	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ¬ Ⅎ 𝑥 𝑦 )
5	4	con2i	⊢ ( Ⅎ 𝑥 𝑦 → ¬ ∀ 𝑥 𝑥 = 𝑦 )
6		nfcvf	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 )
7	5 6	impbii	⊢ ( Ⅎ 𝑥 𝑦 ↔ ¬ ∀ 𝑥 𝑥 = 𝑦 )