Theorem an72ds 32474

Description: Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)

Hypothesis

Ref	Expression
an72ds.1	⊢ (((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜎)

Assertion

Ref	Expression
an72ds	⊢ (((((((𝜑 ∧ 𝜁) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜎)

Proof of Theorem an72ds

Step	Hyp	Ref	Expression
1		an32 645	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜁) ↔ ((𝜑 ∧ 𝜁) ∧ 𝜓))
2	1	anbi1i 623	. . . . 5 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜁) ∧ 𝜃) ↔ (((𝜑 ∧ 𝜁) ∧ 𝜓) ∧ 𝜃))
3	2	anbi1i 623	. . . 4 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜁) ∧ 𝜃) ∧ 𝜏) ↔ ((((𝜑 ∧ 𝜁) ∧ 𝜓) ∧ 𝜃) ∧ 𝜏))
4	3	anbi1i 623	. . 3 ⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜁) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ↔ (((((𝜑 ∧ 𝜁) ∧ 𝜓) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂))
5		an72ds.1	. . . 4 ⊢ (((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜎)
6	5	an62ds 32473	. . 3 ⊢ (((((((𝜑 ∧ 𝜓) ∧ 𝜁) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜎)
7	4, 6	sylanbr 581	. 2 ⊢ (((((((𝜑 ∧ 𝜁) ∧ 𝜓) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜎)
8	7	an62ds 32473	1 ⊢ (((((((𝜑 ∧ 𝜁) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜎)

Syntax hints:   → wi 4   ∧ wa 395

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-an 396

This theorem is referenced by:  an82ds  32475  1arithufdlem3  33531