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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-mo2tf | Structured version Visualization version GIF version |
Description: Closed form of mof 2551 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 20-Sep-2020.) |
Ref | Expression |
---|---|
wl-mo2tf | ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnae 2427 | . . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
2 | nfa1 2140 | . . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑 | |
3 | 1, 2 | nfan 1894 | . 2 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) |
4 | nfnae 2427 | . . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
5 | nfnf1 2143 | . . . 4 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑 | |
6 | 5 | nfal 2310 | . . 3 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑 |
7 | 4, 6 | nfan 1894 | . 2 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) |
8 | simpl 482 | . 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦) | |
9 | sp 2168 | . . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑) | |
10 | 9 | adantl 481 | . 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → Ⅎ𝑦𝜑) |
11 | 3, 7, 8, 10 | wl-mo2df 36945 | 1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 ∃*wmo 2526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2365 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-mo 2528 |
This theorem is referenced by: (None) |
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