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Metamath Proof Explorer
Description: Distribution of implication with conjunction. (Contributed by NM, 31-May-1999) (Proof shortened by Wolf Lammen, 6-Dec-2012)
|
|
Ref |
Expression |
|
Assertion |
imdistan |
⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜒 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pm5.42 |
⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜒 ) ) ) ) |
| 2 |
|
impexp |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜒 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜒 ) ) ) ) |
| 3 |
1 2
|
bitr4i |
⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜒 ) ) ) |