Theorem an42ds 32459

Description: Inference exchanging the last antecedent with the second one. See also an32s 652. (Contributed by Thierry Arnoux, 3-Jun-2025.)

Hypothesis

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

Assertion

Ref	Expression
an42ds	⊢ ((((𝜑 ∧ 𝜃) ∧ 𝜒) ∧ 𝜓) → 𝜏)

Proof of Theorem an42ds

Step	Hyp	Ref	Expression
1		an32 646	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜃) ∧ 𝜓))
2	1	anbi1i 624	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜃) ∧ 𝜒) ↔ (((𝜑 ∧ 𝜃) ∧ 𝜓) ∧ 𝜒))
3		an32 646	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜃) ∧ 𝜒) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃))
4		an32 646	. . 3 ⊢ ((((𝜑 ∧ 𝜃) ∧ 𝜓) ∧ 𝜒) ↔ (((𝜑 ∧ 𝜃) ∧ 𝜒) ∧ 𝜓))
5	2, 3, 4	3bitr3i 301	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ↔ (((𝜑 ∧ 𝜃) ∧ 𝜒) ∧ 𝜓))
6		an42ds.1	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
7	5, 6	sylbir 235	1 ⊢ ((((𝜑 ∧ 𝜃) ∧ 𝜒) ∧ 𝜓) → 𝜏)

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:  an52ds  32460  chnind  32991  mndlrinvb  33020  fldextrspunlsplem  33716