Theorem antnestALT 36199

Description: Alternative proof of antnest 36194 from the valid schema ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) using laws of nested antecedents. Our proof uses only the laws antnestlaw1 36196 and antnestlaw3 36198. (Contributed by Adrian Ducourtial, 5-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
antnestALT	⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)

Proof of Theorem antnestALT

Step	Hyp	Ref	Expression
1		pm2.27 43	. . . 4 ⊢ (⊤ → ((⊤ → 𝜑) → 𝜑))
2		pm2.27 43	. . . 4 ⊢ (((⊤ → 𝜑) → 𝜑) → ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓))
3	1, 2	syl 18	. . 3 ⊢ (⊤ → ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓))
4	3	mptru 1577	. 2 ⊢ ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓)
5		antnestlaw3 36198	. . . 4 ⊢ (((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) ↔ ((((⊤ → 𝜑) → 𝜓) → 𝜑) → 𝜑))
6		antnestlaw1 36196	. . . . . 6 ⊢ (((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) ↔ ((⊤ → 𝜑) → 𝜓))
7	6	imbi1i 352	. . . . 5 ⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) → 𝜑) ↔ (((⊤ → 𝜑) → 𝜓) → 𝜑))
8	7	imbi1i 352	. . . 4 ⊢ (((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) → 𝜑) → 𝜑) ↔ ((((⊤ → 𝜑) → 𝜓) → 𝜑) → 𝜑))
9	5, 8	bitr4i 281	. . 3 ⊢ (((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) ↔ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) → 𝜑) → 𝜑))
10		antnestlaw3 36198	. . 3 ⊢ (((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) → 𝜑) → 𝜑) ↔ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓))
11	9, 10	bitri 278	. 2 ⊢ (((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) ↔ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓))
12	4, 11	mpbi 233	1 ⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)

This proof depends on syntax axioms:   → wi 4  ⊤wtru 1571

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  df-tru 1573

This theorem is used by: (None)