Theorem 4anpull2 1382

Description: An equivalence of two four-terms conjunctions with the terms regrouped (here, the second sub-conjunct of the first term is pulled separately). (Contributed by Zhi Wang, 4-Sep-2024.) (Proof shortened by Garrett Katz, 26-Jun-2026.)

Assertion

Ref	Expression
4anpull2	⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))

Proof of Theorem 4anpull2

Step	Hyp	Ref	Expression
1		an42 670	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
2		3an4anass 1122	. 2 ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
3	1, 2	bitr4i 281	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103

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  df-3an 1105

This theorem is used by:  evenwodadd  47662  iscnrm3lem4  49773