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Mirrors > Home > NFE Home > Th. List > ordir | GIF version |
Description: Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
Ref | Expression |
---|---|
ordir | ⊢ (((φ ∧ ψ) ∨ χ) ↔ ((φ ∨ χ) ∧ (ψ ∨ χ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordi 834 | . 2 ⊢ ((χ ∨ (φ ∧ ψ)) ↔ ((χ ∨ φ) ∧ (χ ∨ ψ))) | |
2 | orcom 376 | . 2 ⊢ (((φ ∧ ψ) ∨ χ) ↔ (χ ∨ (φ ∧ ψ))) | |
3 | orcom 376 | . . 3 ⊢ ((φ ∨ χ) ↔ (χ ∨ φ)) | |
4 | orcom 376 | . . 3 ⊢ ((ψ ∨ χ) ↔ (χ ∨ ψ)) | |
5 | 3, 4 | anbi12i 678 | . 2 ⊢ (((φ ∨ χ) ∧ (ψ ∨ χ)) ↔ ((χ ∨ φ) ∧ (χ ∨ ψ))) |
6 | 1, 2, 5 | 3bitr4i 268 | 1 ⊢ (((φ ∧ ψ) ∨ χ) ↔ ((φ ∨ χ) ∧ (ψ ∨ χ))) |
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: orddi 839 pm5.62 889 dn1 932 cadan 1392 nchoicelem9 6298 |
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