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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege68a | Structured version Visualization version GIF version |
Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege68a | ⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege57aid 39007 | . 2 ⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → (𝜓 ∧ 𝜒))) | |
2 | frege67a 39020 | . 2 ⊢ ((((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → (𝜓 ∧ 𝜒))) → (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 if-wif 1091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 38925 ax-frege2 38926 ax-frege8 38944 ax-frege52a 38992 ax-frege58a 39010 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-tru 1662 df-fal 1672 |
This theorem is referenced by: (None) |
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