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Mirrors > Home > MPE Home > Th. List > axc16nf | Structured version Visualization version GIF version |
Description: If dtru 5128 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 2094. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2080. (Revised by Wolf lammen, 12-Oct-2021.) |
Ref | Expression |
---|---|
axc16nf | ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1744 | . . . 4 ⊢ (∃𝑧𝜑 ↔ ¬ ∀𝑧 ¬ 𝜑) | |
2 | axc16g 2188 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑧 ¬ 𝜑)) | |
3 | 2 | con1d 142 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑧 ¬ 𝜑 → 𝜑)) |
4 | 1, 3 | syl5bi 234 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → 𝜑)) |
5 | axc16g 2188 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | |
6 | 4, 5 | syld 47 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → ∀𝑧𝜑)) |
7 | 6 | nfd 1754 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1506 ∃wex 1743 Ⅎwnf 1747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-12 2107 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1744 df-nf 1748 |
This theorem is referenced by: nfsb 2490 nfsbd 2492 exists2 2703 |
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