New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > anddi | GIF version |
Description: Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
Ref | Expression |
---|---|
anddi | ⊢ (((φ ∨ ψ) ∧ (χ ∨ θ)) ↔ (((φ ∧ χ) ∨ (φ ∧ θ)) ∨ ((ψ ∧ χ) ∨ (ψ ∧ θ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | andir 838 | . 2 ⊢ (((φ ∨ ψ) ∧ (χ ∨ θ)) ↔ ((φ ∧ (χ ∨ θ)) ∨ (ψ ∧ (χ ∨ θ)))) | |
2 | andi 837 | . . 3 ⊢ ((φ ∧ (χ ∨ θ)) ↔ ((φ ∧ χ) ∨ (φ ∧ θ))) | |
3 | andi 837 | . . 3 ⊢ ((ψ ∧ (χ ∨ θ)) ↔ ((ψ ∧ χ) ∨ (ψ ∧ θ))) | |
4 | 2, 3 | orbi12i 507 | . 2 ⊢ (((φ ∧ (χ ∨ θ)) ∨ (ψ ∧ (χ ∨ θ))) ↔ (((φ ∧ χ) ∨ (φ ∧ θ)) ∨ ((ψ ∧ χ) ∨ (ψ ∧ θ)))) |
5 | 1, 4 | bitri 240 | 1 ⊢ (((φ ∨ ψ) ∧ (χ ∨ θ)) ↔ (((φ ∧ χ) ∨ (φ ∧ θ)) ∨ ((ψ ∧ χ) ∨ (ψ ∧ θ)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∨ wo 357 ∧ 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-or 359 df-an 360 |
This theorem is referenced by: funun 5146 |
Copyright terms: Public domain | W3C validator |