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Metamath Proof Explorer
Description: Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008) (Proof shortened by Wolf Lammen, 24-Nov-2019)
|
|
Ref |
Expression |
|
Hypothesis |
necon4abid.1 |
⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ ¬ 𝜓 ) ) |
|
Assertion |
necon4abid |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝜓 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
necon4abid.1 |
⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ ¬ 𝜓 ) ) |
| 2 |
|
notnotb |
⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 ) |
| 3 |
1
|
necon1bbid |
⊢ ( 𝜑 → ( ¬ ¬ 𝜓 ↔ 𝐴 = 𝐵 ) ) |
| 4 |
2 3
|
bitr2id |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ 𝜓 ) ) |