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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.) Usage of this theorem is discouraged because it depends on ax-13 2366. (New usage is discouraged.) |
Ref | Expression |
---|---|
nfcvb | ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnid 5369 | . . . 4 ⊢ ¬ Ⅎ𝑦𝑦 | |
2 | eqidd 2728 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑦) | |
3 | 2 | drnfc1 2917 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑦)) |
4 | 1, 3 | mtbiri 327 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ Ⅎ𝑥𝑦) |
5 | 4 | con2i 139 | . 2 ⊢ (Ⅎ𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) |
6 | nfcvf 2927 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
7 | 5, 6 | impbii 208 | 1 ⊢ (Ⅎ𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1532 Ⅎwnfc 2878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-13 2366 ax-ext 2698 ax-nul 5300 ax-pow 5359 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-cleq 2719 df-nfc 2880 |
This theorem is referenced by: (None) |
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