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Mirrors > Home > ILE Home > Th. List > ordi | GIF version |
Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
ordi | ⊢ ((φ ∨ (ψ ∧ χ)) ↔ ((φ ∨ ψ) ∧ (φ ∨ χ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 102 | . . . 4 ⊢ ((ψ ∧ χ) → ψ) | |
2 | 1 | orim2i 677 | . . 3 ⊢ ((φ ∨ (ψ ∧ χ)) → (φ ∨ ψ)) |
3 | simpr 103 | . . . 4 ⊢ ((ψ ∧ χ) → χ) | |
4 | 3 | orim2i 677 | . . 3 ⊢ ((φ ∨ (ψ ∧ χ)) → (φ ∨ χ)) |
5 | 2, 4 | jca 290 | . 2 ⊢ ((φ ∨ (ψ ∧ χ)) → ((φ ∨ ψ) ∧ (φ ∨ χ))) |
6 | orc 632 | . . . 4 ⊢ (φ → (φ ∨ (ψ ∧ χ))) | |
7 | 6 | adantl 262 | . . 3 ⊢ (((φ ∨ ψ) ∧ φ) → (φ ∨ (ψ ∧ χ))) |
8 | 6 | adantr 261 | . . . 4 ⊢ ((φ ∧ χ) → (φ ∨ (ψ ∧ χ))) |
9 | olc 631 | . . . 4 ⊢ ((ψ ∧ χ) → (φ ∨ (ψ ∧ χ))) | |
10 | 8, 9 | jaoian 708 | . . 3 ⊢ (((φ ∨ ψ) ∧ χ) → (φ ∨ (ψ ∧ χ))) |
11 | 7, 10 | jaodan 709 | . 2 ⊢ (((φ ∨ ψ) ∧ (φ ∨ χ)) → (φ ∨ (ψ ∧ χ))) |
12 | 5, 11 | impbii 117 | 1 ⊢ ((φ ∨ (ψ ∧ χ)) ↔ ((φ ∨ ψ) ∧ (φ ∨ χ))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∨ wo 628 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: ordir 729 orddi 732 pm5.63dc 852 pm4.43 855 orbididc 859 undi 3179 undif4 3278 elnn1uz2 8320 |
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