Theorem rp-fakeimass 41119

Description: A special case where implication appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)

Assertion

Ref	Expression
rp-fakeimass	⊢ ((𝜑 ∨ 𝜒) ↔ (((𝜑 → 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))))

Proof of Theorem rp-fakeimass

Step	Hyp	Ref	Expression
1		conax1 170	. . . . . . 7 ⊢ (¬ (𝜑 → 𝜓) → ¬ 𝜓)
2	1	pm2.21d 121	. . . . . 6 ⊢ (¬ (𝜑 → 𝜓) → (𝜓 → 𝜒))
3	2	a1d 25	. . . . 5 ⊢ (¬ (𝜑 → 𝜓) → (𝜑 → (𝜓 → 𝜒)))
4		ax-1 6	. . . . . 6 ⊢ (𝜒 → (𝜓 → 𝜒))
5	4	a1d 25	. . . . 5 ⊢ (𝜒 → (𝜑 → (𝜓 → 𝜒)))
6	3, 5	ja 186	. . . 4 ⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))
7		ax-2 7	. . . . 5 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
8	7	com3r 87	. . . 4 ⊢ (𝜑 → ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → 𝜒)))
9	6, 8	impbid2 225	. . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))))
10		ax-1 6	. . . 4 ⊢ (𝜒 → ((𝜑 → 𝜓) → 𝜒))
11	10, 5	2thd 264	. . 3 ⊢ (𝜒 → (((𝜑 → 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))))
12	9, 11	jaoi 854	. 2 ⊢ ((𝜑 ∨ 𝜒) → (((𝜑 → 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))))
13		jarl 125	. . . . 5 ⊢ (((𝜑 → 𝜓) → 𝜒) → (¬ 𝜑 → 𝜒))
14	13	orrd 860	. . . 4 ⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜑 ∨ 𝜒))
15	14	a1d 25	. . 3 ⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 ∨ 𝜒)))
16		simplim 167	. . . . 5 ⊢ (¬ (𝜑 → (𝜓 → 𝜒)) → 𝜑)
17	16	orcd 870	. . . 4 ⊢ (¬ (𝜑 → (𝜓 → 𝜒)) → (𝜑 ∨ 𝜒))
18	17	a1i 11	. . 3 ⊢ (¬ ((𝜑 → 𝜓) → 𝜒) → (¬ (𝜑 → (𝜓 → 𝜒)) → (𝜑 ∨ 𝜒)))
19	15, 18	bija 382	. 2 ⊢ ((((𝜑 → 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))) → (𝜑 ∨ 𝜒))
20	12, 19	impbii 208	1 ⊢ ((𝜑 ∨ 𝜒) ↔ (((𝜑 → 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 844

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 845

This theorem is referenced by: (None)