Metamath Proof Explorer
Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007) (Proof shortened by Andrew Salmon, 25-May-2011)
|
|
Ref |
Expression |
|
Hypothesis |
necon3ai.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
|
Assertion |
necon3ai |
⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝜑 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
necon3ai.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
2 |
|
nne |
⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 ) |
3 |
1 2
|
sylibr |
⊢ ( 𝜑 → ¬ 𝐴 ≠ 𝐵 ) |
4 |
3
|
con2i |
⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝜑 ) |