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Mirrors > Home > MPE Home > Th. List > axc14 | Structured version Visualization version GIF version |
Description: Axiom ax-c14 38490 is redundant if we assume ax-5 1905.
Remark 9.6 in
[Megill] p. 448 (p. 16 of the preprint),
regarding axiom scheme C14'.
Note that 𝑤 is a dummy variable introduced in the proof. Its purpose is to satisfy the distinct variable requirements of dveel2 2455 and ax-5 1905. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc14 | ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbn1 2130 | . . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 ¬ ∀𝑧 𝑧 = 𝑦) | |
2 | dveel2 2455 | . . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑤 ∈ 𝑦 → ∀𝑧 𝑤 ∈ 𝑦)) | |
3 | 1, 2 | hbim1 2286 | . . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑤 ∈ 𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦 → 𝑤 ∈ 𝑦)) |
4 | elequ1 2105 | . . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)) | |
5 | 4 | imbi2d 339 | . . . 4 ⊢ (𝑤 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑤 ∈ 𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦))) |
6 | 3, 5 | dvelim 2444 | . . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦))) |
7 | nfa1 2140 | . . . . 5 ⊢ Ⅎ𝑧∀𝑧 𝑧 = 𝑦 | |
8 | 7 | nfn 1852 | . . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦 |
9 | 8 | 19.21 2195 | . . 3 ⊢ (∀𝑧(¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)) |
10 | 6, 9 | imbitrdi 250 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))) |
11 | 10 | pm2.86d 108 | 1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 |
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-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-13 2365 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 |
This theorem is referenced by: (None) |
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