Description: A more efficient proof of Theorem 9.20 of Clemente p. 45. Compare with ex-natded9.20 . (Contributed by David A. Wheeler, 19-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ex-natded9.20.1 | ⊢ ( 𝜑 → ( 𝜓 ∧ ( 𝜒 ∨ 𝜃 ) ) ) | |
| Assertion | ex-natded9.20-2 | ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜃 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ex-natded9.20.1 | ⊢ ( 𝜑 → ( 𝜓 ∧ ( 𝜒 ∨ 𝜃 ) ) ) | |
| 2 | 1 | simpld | ⊢ ( 𝜑 → 𝜓 ) |
| 3 | 2 | anim1i | ⊢ ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 ∧ 𝜒 ) ) |
| 4 | 3 | orcd | ⊢ ( ( 𝜑 ∧ 𝜒 ) → ( ( 𝜓 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜃 ) ) ) |
| 5 | 2 | anim1i | ⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 ∧ 𝜃 ) ) |
| 6 | 5 | olcd | ⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜃 ) ) ) |
| 7 | 1 | simprd | ⊢ ( 𝜑 → ( 𝜒 ∨ 𝜃 ) ) |
| 8 | 4 6 7 | mpjaodan | ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜃 ) ) ) |