Theorem antnestALT 35688

Description: Alternative proof of antnest 35683 from the valid schema ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) using laws of nested antecedents. Our proof uses only the laws antnestlaw1 35685 and antnestlaw3 35687. (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 42	. . . 4 ⊢ (⊤ → ((⊤ → 𝜑) → 𝜑))
2		pm2.27 42	. . . 4 ⊢ (((⊤ → 𝜑) → 𝜑) → ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓))
3	1, 2	syl 17	. . 3 ⊢ (⊤ → ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓))
4	3	mptru 1547	. 2 ⊢ ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓)
5		antnestlaw3 35687	. . . 4 ⊢ (((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) ↔ ((((⊤ → 𝜑) → 𝜓) → 𝜑) → 𝜑))
6		antnestlaw1 35685	. . . . . 6 ⊢ (((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) ↔ ((⊤ → 𝜑) → 𝜓))
7	6	imbi1i 349	. . . . 5 ⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) → 𝜑) ↔ (((⊤ → 𝜑) → 𝜓) → 𝜑))
8	7	imbi1i 349	. . . 4 ⊢ (((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) → 𝜑) → 𝜑) ↔ ((((⊤ → 𝜑) → 𝜓) → 𝜑) → 𝜑))
9	5, 8	bitr4i 278	. . 3 ⊢ (((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) ↔ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) → 𝜑) → 𝜑))
10		antnestlaw3 35687	. . 3 ⊢ (((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜓) → 𝜑) → 𝜑) ↔ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓))
11	9, 10	bitri 275	. 2 ⊢ (((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) ↔ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓))
12	4, 11	mpbi 230	1 ⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)

Syntax hints:   → wi 4  ⊤wtru 1541

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 207  df-tru 1543

This theorem is referenced by: (None)