Theorem 3ornot23 42018

Description: If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 42356. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
3ornot23	⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))

Proof of Theorem 3ornot23

Step	Hyp	Ref	Expression
1		idd 24	. . 3 ⊢ (¬ 𝜑 → (𝜒 → 𝜒))
2		pm2.21 123	. . 3 ⊢ (¬ 𝜑 → (𝜑 → 𝜒))
3		pm2.21 123	. . 3 ⊢ (¬ 𝜓 → (𝜓 → 𝜒))
4	1, 2, 3	3jaao 1431	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))
5	4	3anidm12 1417	1 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ w3o 1084

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  df-3or 1086  df-3an 1087

This theorem is referenced by:  tratrb  42045  tratrbVD  42370