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Metamath Proof Explorer
Description: Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024) (Proof shortened by Wolf Lammen, 7-May-2025)
|
|
Ref |
Expression |
|
Hypotheses |
anbiim.1 |
⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) ) |
|
|
anbiim.2 |
⊢ ( 𝜓 → ( 𝜃 → 𝜒 ) ) |
|
Assertion |
anbiim |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
anbiim.1 |
⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) ) |
| 2 |
|
anbiim.2 |
⊢ ( 𝜓 → ( 𝜃 → 𝜒 ) ) |
| 3 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) ) |
| 4 |
2
|
adantl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜃 → 𝜒 ) ) |
| 5 |
3 4
|
impbid |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) ) |