Metamath Proof Explorer
Description: Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996) (Proof shortened by Wolf Lammen, 19-May-2013)
|
|
Ref |
Expression |
|
Hypotheses |
pm5.21ni.1 |
⊢ ( 𝜑 → 𝜓 ) |
|
|
pm5.21ni.2 |
⊢ ( 𝜒 → 𝜓 ) |
|
Assertion |
pm5.21ni |
⊢ ( ¬ 𝜓 → ( 𝜑 ↔ 𝜒 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
pm5.21ni.1 |
⊢ ( 𝜑 → 𝜓 ) |
2 |
|
pm5.21ni.2 |
⊢ ( 𝜒 → 𝜓 ) |
3 |
1
|
con3i |
⊢ ( ¬ 𝜓 → ¬ 𝜑 ) |
4 |
2
|
con3i |
⊢ ( ¬ 𝜓 → ¬ 𝜒 ) |
5 |
3 4
|
2falsed |
⊢ ( ¬ 𝜓 → ( 𝜑 ↔ 𝜒 ) ) |