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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege5 | Structured version Visualization version GIF version |
Description: A closed form of syl 17. Identical to imim2 58. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege5 | ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege1 42843 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) | |
2 | frege4 42852 | . 2 ⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-frege1 42843 ax-frege2 42844 |
This theorem is referenced by: rp-frege25 42858 frege6 42859 frege7 42861 frege9 42865 frege12 42866 frege16 42869 frege25 42870 frege18 42871 frege22 42872 frege14 42876 frege29 42884 frege34 42890 frege45 42902 frege80 42996 frege90 43006 |
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