Theorem anandi 590

Description: Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)

Assertion

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

Proof of Theorem anandi

Step	Hyp	Ref	Expression
1		anidm 396	. . 3 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
2	1	anbi1i 458	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ (𝜓 ∧ 𝜒)) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
3		an4 586	. 2 ⊢ (((𝜑 ∧ 𝜑) ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
4	2, 3	bitr3i 186	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))

Syntax hints:   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  anandi3  993  moanim  2119  difundi  3415  inrab  3435  uniin  3859  xpcom  5216  fin  5444  fndmin  5669  nnaord  6567  ixpin  6782  ltexprlemdisj  7673  bldisj  14637  blininf  14660  lgsquadlem3  15320