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Mirrors > Home > ILE Home > Th. List > con1dc | GIF version |
Description: Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.) |
Ref | Expression |
---|---|
con1dc | ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnot 591 | . . 3 ⊢ (𝜓 → ¬ ¬ 𝜓) | |
2 | 1 | imim2i 12 | . 2 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜑 → ¬ ¬ 𝜓)) |
3 | condc 782 | . 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → 𝜑))) | |
4 | 2, 3 | syl5 32 | 1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 775 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-dc 776 |
This theorem is referenced by: impidc 788 simplimdc 790 con1biimdc 800 con1bdc 805 pm3.13dc 900 necon1aidc 2296 necon1bidc 2297 necon1addc 2321 necon1bddc 2322 |
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