Metamath Proof Explorer
Description: Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007) (Proof shortened by Wolf Lammen, 22-Nov-2019)
|
|
Ref |
Expression |
|
Hypothesis |
necon1ai.1 |
⊢ ( ¬ 𝜑 → 𝐴 = 𝐵 ) |
|
Assertion |
necon1ai |
⊢ ( 𝐴 ≠ 𝐵 → 𝜑 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
necon1ai.1 |
⊢ ( ¬ 𝜑 → 𝐴 = 𝐵 ) |
2 |
1
|
necon3ai |
⊢ ( 𝐴 ≠ 𝐵 → ¬ ¬ 𝜑 ) |
3 |
2
|
notnotrd |
⊢ ( 𝐴 ≠ 𝐵 → 𝜑 ) |