Theorem an42ds

Description: Inference exchanging the last antecedent with the second one. See also an32s . (Contributed by Thierry Arnoux, 3-Jun-2025)

	Ref	Expression
Hypothesis	an42ds.1	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
Assertion	an42ds	⊢ ( ( ( ( 𝜑 ∧ 𝜃 ) ∧ 𝜒 ) ∧ 𝜓 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		an42ds.1	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
2		an32	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) ↔ ( ( 𝜑 ∧ 𝜃 ) ∧ 𝜓 ) )
3	2	anbi1i	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) ∧ 𝜒 ) ↔ ( ( ( 𝜑 ∧ 𝜃 ) ∧ 𝜓 ) ∧ 𝜒 ) )
4		an32	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) ∧ 𝜒 ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) )
5		an32	⊢ ( ( ( ( 𝜑 ∧ 𝜃 ) ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( ( ( 𝜑 ∧ 𝜃 ) ∧ 𝜒 ) ∧ 𝜓 ) )
6	3 4 5	3bitr3i	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) ↔ ( ( ( 𝜑 ∧ 𝜃 ) ∧ 𝜒 ) ∧ 𝜓 ) )
7	6 1	sylbir	⊢ ( ( ( ( 𝜑 ∧ 𝜃 ) ∧ 𝜒 ) ∧ 𝜓 ) → 𝜏 )