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Mirrors > Home > MPE Home > Th. List > axc16nf | Structured version Visualization version GIF version |
Description: If dtru 5429 is false, then there is only one element in the universe, so everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2146. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2129. (Revised by Wolf Lammen, 12-Oct-2021.) |
Ref | Expression |
---|---|
axc16nf | ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc16g 2243 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑧 ¬ 𝜑)) | |
2 | eximal 1776 | . . . 4 ⊢ ((∃𝑧𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑧 ¬ 𝜑)) | |
3 | 1, 2 | sylibr 233 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → 𝜑)) |
4 | axc16g 2243 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | |
5 | 3, 4 | syld 47 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → ∀𝑧𝜑)) |
6 | 5 | nfd 1784 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 |
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-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 |
This theorem is referenced by: nfsbd 2515 exists2 2651 |
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