Theorem orim12dALT 908

Description: Alternate proof of orim12d 961 which does not depend on df-an 399. This is an illustration of the conservativity of definitions (definitions do not permit to prove additional theorems whose statements do not contain the defined symbol). (Contributed by Wolf Lammen, 8-Aug-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
orim12dALT.1	⊢ (𝜑 → (𝜓 → 𝜒))
orim12dALT.2	⊢ (𝜑 → (𝜃 → 𝜏))

Assertion

Ref	Expression
orim12dALT	⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏)))

Proof of Theorem orim12dALT

Step	Hyp	Ref	Expression
1		pm2.53 847	. 2 ⊢ ((𝜓 ∨ 𝜃) → (¬ 𝜓 → 𝜃))
2		orim12dALT.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	con3d 155	. . 3 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
4		orim12dALT.2	. . 3 ⊢ (𝜑 → (𝜃 → 𝜏))
5	3, 4	imim12d 81	. 2 ⊢ (𝜑 → ((¬ 𝜓 → 𝜃) → (¬ 𝜒 → 𝜏)))
6		pm2.54 848	. 2 ⊢ ((¬ 𝜒 → 𝜏) → (𝜒 ∨ 𝜏))
7	1, 5, 6	syl56 36	1 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 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 209  df-or 844

This theorem is referenced by: (None)