Theorem bnj257 32586

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 460	. . 3 ⊢ ((𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜒))
2	1	anbi2i 622	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜒)))
3		bnj256 32585	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
4		bnj256 32585	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜒)))
5	2, 3, 4	3bitr4i 302	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ 𝜃 ∧ 𝜒))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w-bnj17 32565

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 206  df-an 396  df-3an 1087  df-bnj17 32566

This theorem is referenced by:  bnj258  32587  bnj334  32592  bnj543  32773  bnj929  32816