Theorem anandir 802

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 625	. . 3 ⊢ ((χ ∧ χ) ↔ χ)
2	1	anbi2i 675	. 2 ⊢ (((φ ∧ ψ) ∧ (χ ∧ χ)) ↔ ((φ ∧ ψ) ∧ χ))
3		an4 797	. 2 ⊢ (((φ ∧ ψ) ∧ (χ ∧ χ)) ↔ ((φ ∧ χ) ∧ (ψ ∧ χ)))
4	2, 3	bitr3i 242	1 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ((φ ∧ χ) ∧ (ψ ∧ χ)))

Syntax hints:   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by:  cadan  1392  fununi  5161  imadif  5172  restxp  5787