This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999) (Proof shortened by Wolf Lammen, 11-Nov-2012)
|
|
Ref |
Expression |
|
Assertion |
biimpr |
⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dfbi1 |
⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) |
| 2 |
|
simprim |
⊢ ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( 𝜓 → 𝜑 ) ) |
| 3 |
1 2
|
sylbi |
⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) ) |