![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dtrucor2 | Structured version Visualization version GIF version |
Description: The theorem form of the deduction dtrucor 5237 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5237. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dtrucor2.1 | ⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦) |
Ref | Expression |
---|---|
dtrucor2 | ⊢ (𝜑 ∧ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2390 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | dtrucor2.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦) | |
3 | 2 | necon2bi 3017 | . . . 4 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 = 𝑦) |
4 | pm2.01 192 | . . . 4 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 = 𝑦) → ¬ 𝑥 = 𝑦) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ¬ 𝑥 = 𝑦 |
6 | 5 | nex 1802 | . 2 ⊢ ¬ ∃𝑥 𝑥 = 𝑦 |
7 | 1, 6 | pm2.24ii 120 | 1 ⊢ (𝜑 ∧ ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∃wex 1781 ≠ wne 2987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-ne 2988 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |