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Metamath Proof Explorer
Description: A contraposition deduction. (Contributed by NM, 21-May-1994)
|
|
Ref |
Expression |
|
Hypothesis |
con4bid.1 |
⊢ ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) |
|
Assertion |
con4bid |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
con4bid.1 |
⊢ ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) |
| 2 |
1
|
biimprd |
⊢ ( 𝜑 → ( ¬ 𝜒 → ¬ 𝜓 ) ) |
| 3 |
2
|
con4d |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
| 4 |
1
|
biimpd |
⊢ ( 𝜑 → ( ¬ 𝜓 → ¬ 𝜒 ) ) |
| 5 |
3 4
|
impcon4bid |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |