Theorem antnestlaw2 36457

Description: A law of nested antecedents. (Contributed by Adrian Ducourtial, 5-Dec-2025.)

Assertion

Ref	Expression
antnestlaw2	⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜒))

Proof of Theorem antnestlaw2

Step	Hyp	Ref	Expression
1		pm2.27 43	. . . . . 6 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2	1	a1d 26	. . . . 5 ⊢ (𝜑 → (¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒) → ((𝜑 → 𝜓) → 𝜓)))
3		pm2.21 124	. . . . . . . 8 ⊢ (¬ 𝜑 → (𝜑 → 𝜒))
4	3	a1d 26	. . . . . . 7 ⊢ (¬ 𝜑 → (¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒) → (𝜑 → 𝜒)))
5		simplim 168	. . . . . . 7 ⊢ (¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒) → ((𝜑 → 𝜒) → 𝜓))
6	4, 5	sylcom 31	. . . . . 6 ⊢ (¬ 𝜑 → (¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒) → 𝜓))
7	6	a1dd 51	. . . . 5 ⊢ (¬ 𝜑 → (¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒) → ((𝜑 → 𝜓) → 𝜓)))
8	2, 7	pm2.61i 184	. . . 4 ⊢ (¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒) → ((𝜑 → 𝜓) → 𝜓))
9		conax1 171	. . . 4 ⊢ (¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒) → ¬ 𝜒)
10	8, 9	jcnd 164	. . 3 ⊢ (¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒) → ¬ (((𝜑 → 𝜓) → 𝜓) → 𝜒))
11	10	con4i 115	. 2 ⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) → (((𝜑 → 𝜒) → 𝜓) → 𝜒))
12		conax1 171	. . . 4 ⊢ (¬ (((𝜑 → 𝜓) → 𝜓) → 𝜒) → ¬ 𝜒)
13		con3 154	. . . . . . . 8 ⊢ ((𝜑 → 𝜒) → (¬ 𝜒 → ¬ 𝜑))
14	12, 13	syl5com 32	. . . . . . 7 ⊢ (¬ (((𝜑 → 𝜓) → 𝜓) → 𝜒) → ((𝜑 → 𝜒) → ¬ 𝜑))
15		pm2.21 124	. . . . . . 7 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
16	14, 15	syl6 36	. . . . . 6 ⊢ (¬ (((𝜑 → 𝜓) → 𝜓) → 𝜒) → ((𝜑 → 𝜒) → (𝜑 → 𝜓)))
17		pm2.521g2 176	. . . . . 6 ⊢ (¬ (((𝜑 → 𝜓) → 𝜓) → 𝜒) → ((𝜑 → 𝜒) → ((𝜑 → 𝜓) → 𝜓)))
18	16, 17	mpdd 44	. . . . 5 ⊢ (¬ (((𝜑 → 𝜓) → 𝜓) → 𝜒) → ((𝜑 → 𝜒) → 𝜓))
19		jcn 163	. . . . . 6 ⊢ (((𝜑 → 𝜒) → 𝜓) → (¬ 𝜒 → ¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒)))
20	19	a1i 11	. . . . 5 ⊢ (¬ (((𝜑 → 𝜓) → 𝜓) → 𝜒) → (((𝜑 → 𝜒) → 𝜓) → (¬ 𝜒 → ¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒))))
21	18, 20	mpd 16	. . . 4 ⊢ (¬ (((𝜑 → 𝜓) → 𝜓) → 𝜒) → (¬ 𝜒 → ¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒)))
22	12, 21	mpd 16	. . 3 ⊢ (¬ (((𝜑 → 𝜓) → 𝜓) → 𝜒) → ¬ (((𝜑 → 𝜒) → 𝜓) → 𝜒))
23	22	con4i 115	. 2 ⊢ ((((𝜑 → 𝜒) → 𝜓) → 𝜒) → (((𝜑 → 𝜓) → 𝜓) → 𝜒))
24	11, 23	impbii 212	1 ⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by: (None)