Theorem andir 457

Description: Distributive law for conjunction.

Assertion

Ref	Expression
andir	⊢ (((φ ∨ ψ) ∧ χ) ↔ ((φ ∧ χ) ∨ (ψ ∧ χ)))

Proof of Theorem andir

Step	Hyp	Ref	Expression
1		andi 456	. 2 ⊢ ((χ ∧ (φ ∨ ψ)) ↔ ((χ ∧ φ) ∨ (χ ∧ ψ)))
2		ancom 333	. 2 ⊢ (((φ ∨ ψ) ∧ χ) ↔ (χ ∧ (φ ∨ ψ)))
3		ancom 333	. . 3 ⊢ ((φ ∧ χ) ↔ (χ ∧ φ))
4		ancom 333	. . 3 ⊢ ((ψ ∧ χ) ↔ (χ ∧ ψ))
5	3, 4	orbi12i 216	. 2 ⊢ (((φ ∧ χ) ∨ (ψ ∧ χ)) ↔ ((χ ∧ φ) ∨ (χ ∧ ψ)))
6	1, 2, 5	3bitr4 158	1 ⊢ (((φ ∨ ψ) ∧ χ) ↔ ((φ ∧ χ) ∨ (ψ ∧ χ)))

Syntax hints:   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  anddi 459  biass 511  caselem 561  iunxun 2035  xpundir 2462  nnmcan 3190

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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