This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Infer implication from a logical equivalence. Similar to biimpar .
(Contributed by NM, 2-Jan-2009)
|
|
Ref |
Expression |
|
Hypothesis |
biimp3a.1 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) ) |
|
Assertion |
biimp3ar |
⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜒 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
biimp3a.1 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) ) |
| 2 |
1
|
exbiri |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) |
| 3 |
2
|
3imp |
⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜒 ) |