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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege33 | Structured version Visualization version GIF version |
Description: If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 146. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege33 | ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege28 42190 | . 2 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) | |
2 | frege32 42195 | . 2 ⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-frege1 42150 ax-frege2 42151 ax-frege28 42190 ax-frege31 42194 |
This theorem is referenced by: frege34 42197 frege46 42210 |
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