Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfcvb | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nfcvb | ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnid 4823 | . . . 4 ⊢ ¬ Ⅎ𝑦𝑦 | |
2 | eqidd 2611 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑦) | |
3 | 2 | drnfc1 2768 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑦)) |
4 | 1, 3 | mtbiri 316 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ Ⅎ𝑥𝑦) |
5 | 4 | con2i 133 | . 2 ⊢ (Ⅎ𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) |
6 | nfcvf 2774 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
7 | 5, 6 | impbii 198 | 1 ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∀wal 1473 Ⅎwnfc 2738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-cleq 2603 df-clel 2606 df-nfc 2740 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |