Theorem bnj257 35331

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

Assertion

Ref	Expression
bnj257	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ 𝜃 ∧ 𝜒))

Proof of Theorem bnj257

Step	Hyp	Ref	Expression
1		ancom 466	. . 3 ⊢ ((𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜒))
2	1	anbi2i 635	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜒)))
3		bnj256 35330	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
4		bnj256 35330	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜒)))
5	2, 3, 4	3bitr4i 306	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ 𝜃 ∧ 𝜒))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w-bnj17 35310

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 35311

This theorem is used by:  bnj258  35332  bnj334  35337  bnj543  35516  bnj929  35559