Theorem ex-natded5.7-2 26661

Description: A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 26660. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ex-natded5.7.1	⊢ (𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃)))

Assertion

Ref	Expression
ex-natded5.7-2	⊢ (𝜑 → (𝜓 ∨ 𝜒))

Proof of Theorem ex-natded5.7-2

Step	Hyp	Ref	Expression
1		ex-natded5.7.1	. 2 ⊢ (𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃)))
2		simpl 472	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜒)
3	2	orim2i 539	. 2 ⊢ ((𝜓 ∨ (𝜒 ∧ 𝜃)) → (𝜓 ∨ 𝜒))
4	1, 3	syl 17	1 ⊢ (𝜑 → (𝜓 ∨ 𝜒))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383

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 196  df-or 384  df-an 385

This theorem is referenced by: (None)