Description: Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | ifpan123g | ⊢ ( ( if- ( 𝜑 , 𝜒 , 𝜏 ) ∧ if- ( 𝜓 , 𝜃 , 𝜂 ) ) ↔ ( ( ( ¬ 𝜑 ∨ 𝜒 ) ∧ ( 𝜑 ∨ 𝜏 ) ) ∧ ( ( ¬ 𝜓 ∨ 𝜃 ) ∧ ( 𝜓 ∨ 𝜂 ) ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfifp4 | ⊢ ( if- ( 𝜑 , 𝜒 , 𝜏 ) ↔ ( ( ¬ 𝜑 ∨ 𝜒 ) ∧ ( 𝜑 ∨ 𝜏 ) ) ) | |
2 | dfifp4 | ⊢ ( if- ( 𝜓 , 𝜃 , 𝜂 ) ↔ ( ( ¬ 𝜓 ∨ 𝜃 ) ∧ ( 𝜓 ∨ 𝜂 ) ) ) | |
3 | 1 2 | anbi12i | ⊢ ( ( if- ( 𝜑 , 𝜒 , 𝜏 ) ∧ if- ( 𝜓 , 𝜃 , 𝜂 ) ) ↔ ( ( ( ¬ 𝜑 ∨ 𝜒 ) ∧ ( 𝜑 ∨ 𝜏 ) ) ∧ ( ( ¬ 𝜓 ∨ 𝜃 ) ∧ ( 𝜓 ∨ 𝜂 ) ) ) ) |