Theorem bnj312 35166

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

Assertion

Ref	Expression
bnj312	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒 ∧ 𝜃))

Proof of Theorem bnj312

Step	Hyp	Ref	Expression
1		3ancoma 1115	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒))
2	1	anbi1i 636	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜓 ∧ 𝜑 ∧ 𝜒) ∧ 𝜃))
3		df-bnj17 35141	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
4		df-bnj17 35141	. 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜑 ∧ 𝜒) ∧ 𝜃))
5	2, 3, 4	3bitr4i 306	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒 ∧ 𝜃))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∧ w-bnj17 35140

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  df-bnj17 35141

This theorem is used by:  bnj334  35167  bnj563  35197  bnj953  35392