Theorem ex-natded5.13-2 28681

Description: A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 28680. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ex-natded5.13.1	⊢ (𝜑 → (𝜓 ∨ 𝜒))
ex-natded5.13.2	⊢ (𝜑 → (𝜓 → 𝜃))
ex-natded5.13.3	⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒))

Assertion

Ref	Expression
ex-natded5.13-2	⊢ (𝜑 → (𝜃 ∨ 𝜏))

Proof of Theorem ex-natded5.13-2

Step	Hyp	Ref	Expression
1		ex-natded5.13.1	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
2		ex-natded5.13.2	. . 3 ⊢ (𝜑 → (𝜓 → 𝜃))
3		ex-natded5.13.3	. . . 4 ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒))
4	3	con4d 115	. . 3 ⊢ (𝜑 → (𝜒 → 𝜏))
5	2, 4	orim12d 961	. 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) → (𝜃 ∨ 𝜏)))
6	1, 5	mpd 15	1 ⊢ (𝜑 → (𝜃 ∨ 𝜏))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 843

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 206  df-an 396  df-or 844

This theorem is referenced by: (None)