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Metamath Proof Explorer
Description: Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007) (Proof shortened by Wolf Lammen, 24-Nov-2019)
|
|
Ref |
Expression |
|
Hypothesis |
necon2abid.1 |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ¬ 𝜓 ) ) |
|
Assertion |
necon2abid |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝐴 ≠ 𝐵 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
necon2abid.1 |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ¬ 𝜓 ) ) |
| 2 |
|
notnotb |
⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 ) |
| 3 |
1
|
necon3abid |
⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓 ) ) |
| 4 |
2 3
|
bitr4id |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝐴 ≠ 𝐵 ) ) |