Theorem an52ds

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

	Ref	Expression
Hypothesis	an52ds.1	⊢ ( ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
Assertion	an52ds	⊢ ( ( ( ( ( 𝜑 ∧ 𝜏 ) ∧ 𝜒 ) ∧ 𝜃 ) ∧ 𝜓 ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		an52ds.1	⊢ ( ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
2		an32	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜏 ) ↔ ( ( 𝜑 ∧ 𝜏 ) ∧ 𝜓 ) )
3	2	anbi1i	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜏 ) ∧ 𝜃 ) ↔ ( ( ( 𝜑 ∧ 𝜏 ) ∧ 𝜓 ) ∧ 𝜃 ) )
4	1	an42ds	⊢ ( ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜏 ) ∧ 𝜃 ) ∧ 𝜒 ) → 𝜂 )
5	3 4	sylanbr	⊢ ( ( ( ( ( 𝜑 ∧ 𝜏 ) ∧ 𝜓 ) ∧ 𝜃 ) ∧ 𝜒 ) → 𝜂 )
6	5	an42ds	⊢ ( ( ( ( ( 𝜑 ∧ 𝜏 ) ∧ 𝜒 ) ∧ 𝜃 ) ∧ 𝜓 ) → 𝜂 )