Theorem ex-natded9.20-2 30437

Description: A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 30436. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ex-natded9.20.1	⊢ (𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃)))

Assertion

Ref	Expression
ex-natded9.20-2	⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)))

Proof of Theorem ex-natded9.20-2

Step	Hyp	Ref	Expression
1		ex-natded9.20.1	. . . . 5 ⊢ (𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃)))
2	1	simpld 494	. . . 4 ⊢ (𝜑 → 𝜓)
3	2	anim1i 615	. . 3 ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒))
4	3	orcd 874	. 2 ⊢ ((𝜑 ∧ 𝜒) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)))
5	2	anim1i 615	. . 3 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ∧ 𝜃))
6	5	olcd 875	. 2 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)))
7	1	simprd 495	. 2 ⊢ (𝜑 → (𝜒 ∨ 𝜃))
8	4, 6, 7	mpjaodan 961	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849

This theorem is referenced by: (None)