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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 5167 | . . . 4 ⊢ ¬ Ⅎ𝑦𝑦 | |
2 | eqidd 2796 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑦) | |
3 | 2 | drnfc1 2966 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑦)) |
4 | 1, 3 | mtbiri 328 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ Ⅎ𝑥𝑦) |
5 | 4 | con2i 141 | . 2 ⊢ (Ⅎ𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) |
6 | nfcvf 2975 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
7 | 5, 6 | impbii 210 | 1 ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∀wal 1520 Ⅎwnfc 2933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-nul 5101 ax-pow 5157 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-cleq 2788 df-clel 2863 df-nfc 2935 |
This theorem is referenced by: (None) |
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