Description: Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994)
Ref | Expression | ||
---|---|---|---|
Hypothesis | pm5.32i.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
Assertion | pm5.32i | ⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.32i.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
2 | pm5.32 | ⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) ) | |
3 | 1 2 | mpbi | ⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) |