This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Wolf Lammen, 23-Nov-2019)
|
|
Ref |
Expression |
|
Hypothesis |
necon1ad.1 |
⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 = 𝐵 ) ) |
|
Assertion |
necon1ad |
⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝜓 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
necon1ad.1 |
⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 = 𝐵 ) ) |
| 2 |
1
|
necon3ad |
⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → ¬ ¬ 𝜓 ) ) |
| 3 |
|
notnotr |
⊢ ( ¬ ¬ 𝜓 → 𝜓 ) |
| 4 |
2 3
|
syl6 |
⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝜓 ) ) |