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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