Theorem an52ds 32873

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 659	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜏) ↔ ((𝜑 ∧ 𝜏) ∧ 𝜓))
2	1	anbi1i 636	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜃) ↔ (((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜃))
3		an52ds.1	. . . 4 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)
4	3	an42ds 1520	. . 3 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜃) ∧ 𝜒) → 𝜂)
5	2, 4	sylanbr 594	. 2 ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜃) ∧ 𝜒) → 𝜂)
6	5	an42ds 1520	1 ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜒) ∧ 𝜃) ∧ 𝜓) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  an62ds  32874  dfufd2  33904  esplyfval1  34027