Theorem an52ds 32472

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

Hypothesis

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

Assertion

Ref	Expression
an52ds	⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜒) ∧ 𝜃) ∧ 𝜓) → 𝜂)

Proof of Theorem an52ds

Step	Hyp	Ref	Expression
1		an32 645	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜏) ↔ ((𝜑 ∧ 𝜏) ∧ 𝜓))
2	1	anbi1i 623	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜃) ↔ (((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜃))
3		an52ds.1	. . . 4 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)
4	3	an42ds 32471	. . 3 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜃) ∧ 𝜒) → 𝜂)
5	2, 4	sylanbr 581	. 2 ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜃) ∧ 𝜒) → 𝜂)
6	5	an42ds 32471	1 ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜒) ∧ 𝜃) ∧ 𝜓) → 𝜂)

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:  an62ds  32473  dfufd2  33535