Theorem anandir 677

Description: Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)

Assertion

Ref	Expression
anandir	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)))

Proof of Theorem anandir

Step	Hyp	Ref	Expression
1		anidm 564	. . 3 ⊢ ((𝜒 ∧ 𝜒) ↔ 𝜒)
2	1	anbi2i 623	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
3		an4 656	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)))
4	2, 3	bitr3i 277	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)))

Syntax hints:   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  anandi3r  1103  disjxun  5141  fununi  6641  imadif  6650  elfzuzb  13558  frgr3v  30294  5oalem3  31675  5oalem5  31677  refrelredund4  38636  nzin  44337  un2122  44810