Description: Distribution of implication over biconditional (inference form). (Contributed by NM, 12-Mar-1995)
Ref | Expression | ||
---|---|---|---|
Hypothesis | pm5.32i.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
Assertion | pm5.32ri | ⊢ ( ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜒 ∧ 𝜑 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.32i.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
2 | 1 | pm5.32i | ⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) |
3 | ancom | ⊢ ( ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) ) | |
4 | ancom | ⊢ ( ( 𝜒 ∧ 𝜑 ) ↔ ( 𝜑 ∧ 𝜒 ) ) | |
5 | 2 3 4 | 3bitr4i | ⊢ ( ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜒 ∧ 𝜑 ) ) |