Theorem bnj250 34999

Description: ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj250	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))

Proof of Theorem bnj250

Step	Hyp	Ref	Expression
1		df-bnj17 34985	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
2		3anass 1107	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
3	2	anbi1i 633	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃))
4		anass 472	. 2 ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
5	1, 3, 4	3bitri 299	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   ∧ w-bnj17 34984

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 209  df-an 400  df-3an 1101  df-bnj17 34985

This theorem is referenced by:  bnj251  35000  bnj252  35001  bnj345  35012