Theorem an62ds 32331

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

Hypothesis

Ref	Expression
an62ds.1	⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
an62ds	⊢ ((((((𝜑 ∧ 𝜂) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜁)

Proof of Theorem an62ds

Step	Hyp	Ref	Expression
1		an32 644	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜂) ↔ ((𝜑 ∧ 𝜂) ∧ 𝜓))
2	1	anbi1i 622	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜂) ∧ 𝜃) ↔ (((𝜑 ∧ 𝜂) ∧ 𝜓) ∧ 𝜃))
3	2	anbi1i 622	. . 3 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜂) ∧ 𝜃) ∧ 𝜏) ↔ ((((𝜑 ∧ 𝜂) ∧ 𝜓) ∧ 𝜃) ∧ 𝜏))
4		an62ds.1	. . . 4 ⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)
5	4	an52ds 32330	. . 3 ⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜂) ∧ 𝜃) ∧ 𝜏) ∧ 𝜒) → 𝜁)
6	3, 5	sylanbr 580	. 2 ⊢ ((((((𝜑 ∧ 𝜂) ∧ 𝜓) ∧ 𝜃) ∧ 𝜏) ∧ 𝜒) → 𝜁)
7	6	an52ds 32330	1 ⊢ ((((((𝜑 ∧ 𝜂) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜁)

Syntax hints:   → wi 4   ∧ wa 394

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-an 395

This theorem is referenced by:  an72ds  32332  dfufd2lem  33364