Theorem an62ds

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

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

Proof

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