Description: Deduction associated with biadani . Add a conjunction to an equivalence. (Contributed by Thierry Arnoux, 16-Jun-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | biadanid.1 | ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) | |
biadanid.2 | ⊢ ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 ↔ 𝜃 ) ) | ||
Assertion | biadanid | ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biadanid.1 | ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) | |
2 | biadanid.2 | ⊢ ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 ↔ 𝜃 ) ) | |
3 | 2 | biimpa | ⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ 𝜓 ) → 𝜃 ) |
4 | 3 | an32s | ⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 ) |
5 | 1 4 | mpdan | ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 ) |
6 | 1 5 | jca | ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ∧ 𝜃 ) ) |
7 | 2 | biimpar | ⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜓 ) |
8 | 7 | anasss | ⊢ ( ( 𝜑 ∧ ( 𝜒 ∧ 𝜃 ) ) → 𝜓 ) |
9 | 6 8 | impbida | ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) ) |