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Metamath Proof Explorer
Description: Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005) (Proof shortened by Wolf Lammen, 21-Dec-2013)
|
|
Ref |
Expression |
|
Assertion |
ibibr |
⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ( 𝜓 ↔ 𝜑 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pm5.501 |
⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) ) |
| 2 |
|
bicom |
⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜓 ↔ 𝜑 ) ) |
| 3 |
1 2
|
bitrdi |
⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜓 ↔ 𝜑 ) ) ) |
| 4 |
3
|
pm5.74i |
⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ( 𝜓 ↔ 𝜑 ) ) ) |