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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 5025 | . . . 4 ⊢ ¬ Ⅎ𝑦𝑦 | |
2 | eqidd 2771 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑦) | |
3 | 2 | drnfc1 2930 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑦)) |
4 | 1, 3 | mtbiri 316 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ Ⅎ𝑥𝑦) |
5 | 4 | con2i 136 | . 2 ⊢ (Ⅎ𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) |
6 | nfcvf 2936 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
7 | 5, 6 | impbii 199 | 1 ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∀wal 1628 Ⅎwnfc 2899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-nul 4920 ax-pow 4971 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-cleq 2763 df-clel 2766 df-nfc 2901 |
This theorem is referenced by: (None) |
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