Metamath Proof Explorer
Description: A variation on impbid with contraposition. (Contributed by Jeff
Hankins, 3-Jul-2009)
|
|
Ref |
Expression |
|
Hypotheses |
impcon4bid.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
|
|
impcon4bid.2 |
⊢ ( 𝜑 → ( ¬ 𝜓 → ¬ 𝜒 ) ) |
|
Assertion |
impcon4bid |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
impcon4bid.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
2 |
|
impcon4bid.2 |
⊢ ( 𝜑 → ( ¬ 𝜓 → ¬ 𝜒 ) ) |
3 |
2
|
con4d |
⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) ) |
4 |
1 3
|
impbid |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |