Theorem ecase23d 1481

Description: Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.)

Hypotheses

Ref	Expression
ecase23d.1	⊢ (𝜑 → ¬ 𝜒)
ecase23d.2	⊢ (𝜑 → ¬ 𝜃)
ecase23d.3	⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))

Assertion

Ref	Expression
ecase23d	⊢ (𝜑 → 𝜓)

Proof of Theorem ecase23d

Step	Hyp	Ref	Expression
1		ecase23d.3	. . 3 ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))
2		3orass 1095	. . 3 ⊢ ((𝜓 ∨ 𝜒 ∨ 𝜃) ↔ (𝜓 ∨ (𝜒 ∨ 𝜃)))
3	1, 2	sylib 219	. 2 ⊢ (𝜑 → (𝜓 ∨ (𝜒 ∨ 𝜃)))
4		ecase23d.1	. . 3 ⊢ (𝜑 → ¬ 𝜒)
5		ecase23d.2	. . 3 ⊢ (𝜑 → ¬ 𝜃)
6		ioran 991	. . 3 ⊢ (¬ (𝜒 ∨ 𝜃) ↔ (¬ 𝜒 ∧ ¬ 𝜃))
7	4, 5, 6	sylanbrc 589	. 2 ⊢ (𝜑 → ¬ (𝜒 ∨ 𝜃))
8	3, 7	olcnd 883	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 853   ∨ w3o 1091

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 208  df-an 397  df-or 854  df-3or 1093

This theorem is referenced by:  tz7.7  6336  chnpof1  18587  nolt02o  27677  nogt01o  27678  noresle  27679  nosupbnd1lem6  27695  nosupbnd2lem1  27697  noinfbnd1lem6  27710  ltmuls2  28181  rexmul2  32846  archiabllem2b  33277