Description: Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011) (Proof shortened by Wolf Lammen, 14-Sep-2013)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | imim21b | ⊢ ( ( 𝜓 → 𝜑 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜓 → 𝜃 ) ) ↔ ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bi2.04 | ⊢ ( ( ( 𝜑 → 𝜒 ) → ( 𝜓 → 𝜃 ) ) ↔ ( 𝜓 → ( ( 𝜑 → 𝜒 ) → 𝜃 ) ) ) | |
| 2 | pm5.5 | ⊢ ( 𝜑 → ( ( 𝜑 → 𝜒 ) ↔ 𝜒 ) ) | |
| 3 | 2 | imbi1d | ⊢ ( 𝜑 → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) ↔ ( 𝜒 → 𝜃 ) ) ) |
| 4 | 3 | imim2i | ⊢ ( ( 𝜓 → 𝜑 ) → ( 𝜓 → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) ↔ ( 𝜒 → 𝜃 ) ) ) ) |
| 5 | 4 | pm5.74d | ⊢ ( ( 𝜓 → 𝜑 ) → ( ( 𝜓 → ( ( 𝜑 → 𝜒 ) → 𝜃 ) ) ↔ ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) ) |
| 6 | 1 5 | syl5bb | ⊢ ( ( 𝜓 → 𝜑 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜓 → 𝜃 ) ) ↔ ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) ) |