Theorem anddi 459

Description: Double distributive law for conjunction.

Assertion

Ref	Expression
anddi	⊢ (((φ ∨ ψ) ∧ (χ ∨ θ)) ↔ (((φ ∧ χ) ∨ (φ ∧ θ)) ∨ ((ψ ∧ χ) ∨ (ψ ∧ θ))))

Proof of Theorem anddi

Step	Hyp	Ref	Expression
1		andir 457	. 2 ⊢ (((φ ∨ ψ) ∧ (χ ∨ θ)) ↔ ((φ ∧ (χ ∨ θ)) ∨ (ψ ∧ (χ ∨ θ))))
2		andi 456	. . 3 ⊢ ((φ ∧ (χ ∨ θ)) ↔ ((φ ∧ χ) ∨ (φ ∧ θ)))
3		andi 456	. . 3 ⊢ ((ψ ∧ (χ ∨ θ)) ↔ ((ψ ∧ χ) ∨ (ψ ∧ θ)))
4	2, 3	orbi12i 216	. 2 ⊢ (((φ ∧ (χ ∨ θ)) ∨ (ψ ∧ (χ ∨ θ))) ↔ (((φ ∧ χ) ∨ (φ ∧ θ)) ∨ ((ψ ∧ χ) ∨ (ψ ∧ θ))))
5	1, 4	bitr 151	1 ⊢ (((φ ∨ ψ) ∧ (χ ∨ θ)) ↔ (((φ ∧ χ) ∨ (φ ∧ θ)) ∨ ((ψ ∧ χ) ∨ (ψ ∧ θ))))

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

This theorem is referenced by:  funun 2700

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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