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Metamath Proof Explorer

Theorem 4anpull2

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)

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

Proof

Step	Hyp	Ref	Expression
1		anass	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) ) )
2		anass	⊢ ( ( ( 𝜑 ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) ↔ ( 𝜑 ∧ ( ( 𝜒 ∧ 𝜃 ) ∧ 𝜓 ) ) )
3		3anass	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) ↔ ( 𝜑 ∧ ( 𝜒 ∧ 𝜃 ) ) )
4	3	anbi1i	⊢ ( ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜓 ) ↔ ( ( 𝜑 ∧ ( 𝜒 ∧ 𝜃 ) ) ∧ 𝜓 ) )
5		ancom	⊢ ( ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜒 ∧ 𝜃 ) ∧ 𝜓 ) )
6	5	anbi2i	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) ) ↔ ( 𝜑 ∧ ( ( 𝜒 ∧ 𝜃 ) ∧ 𝜓 ) ) )
7	2 4 6	3bitr4ri	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) ) ↔ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜓 ) )
8	1 7	bitri	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜓 ) )