Theorem 3an6 1445

Description: Analogue of an4 653 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
3an6	⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ (𝜓 ∧ 𝜃 ∧ 𝜂)))

Proof of Theorem 3an6

Step	Hyp	Ref	Expression
1		an6 1444	. 2 ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ (𝜓 ∧ 𝜃 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)))
2	1	bicomi 223	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ (𝜓 ∧ 𝜃 ∧ 𝜂)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086

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 397  df-3an 1088

This theorem is referenced by:  an33rean  1482  an33reanOLD  1483  f13dfv  7146  poxp  7969  wfrlem4OLD  8143  cotr2g  14687  axcontlem8  27339  cplgr3v  27802  cgr3tr4  34354