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Mirrors > Home > ILE Home > Th. List > imordc | GIF version |
Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 796, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.) |
Ref | Expression |
---|---|
imordc | ⊢ (DECID φ → ((φ → ψ) ↔ (¬ φ ∨ ψ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotdc 765 | . . 3 ⊢ (DECID φ → (φ ↔ ¬ ¬ φ)) | |
2 | 1 | imbi1d 220 | . 2 ⊢ (DECID φ → ((φ → ψ) ↔ (¬ ¬ φ → ψ))) |
3 | dcn 745 | . . 3 ⊢ (DECID φ → DECID ¬ φ) | |
4 | dfordc 790 | . . 3 ⊢ (DECID ¬ φ → ((¬ φ ∨ ψ) ↔ (¬ ¬ φ → ψ))) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (DECID φ → ((¬ φ ∨ ψ) ↔ (¬ ¬ φ → ψ))) |
6 | 2, 5 | bitr4d 180 | 1 ⊢ (DECID φ → ((φ → ψ) ↔ (¬ φ ∨ ψ))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∨ wo 628 DECID wdc 741 |
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-in1 544 ax-in2 545 ax-io 629 |
This theorem depends on definitions: df-bi 110 df-dc 742 |
This theorem is referenced by: pm4.62dc 797 pm2.26dc 812 nf4dc 1557 |
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