Theorem ecase3d 1030

Description: Deduction for elimination by cases. (Contributed by NM, 2-May-1996.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Hypotheses

Ref	Expression
ecase3d.1	⊢ (𝜑 → (𝜓 → 𝜃))
ecase3d.2	⊢ (𝜑 → (𝜒 → 𝜃))
ecase3d.3	⊢ (𝜑 → (¬ (𝜓 ∨ 𝜒) → 𝜃))

Assertion

Ref	Expression
ecase3d	⊢ (𝜑 → 𝜃)

Proof of Theorem ecase3d

Step	Hyp	Ref	Expression
1		ecase3d.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜃))
2		ecase3d.2	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
3	1, 2	jaod 855	. 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜒) → 𝜃))
4		ecase3d.3	. 2 ⊢ (𝜑 → (¬ (𝜓 ∨ 𝜒) → 𝜃))
5	3, 4	pm2.61d 179	1 ⊢ (𝜑 → 𝜃)

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-or 844

This theorem is referenced by:  ecased  1031  distrlem4pr  10766  lcmdvds  16294  atcvat4i  30738  cvrat4  37436  metakunt13  40117