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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 42232 | . 2 ⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → (𝜓 ∧ 𝜒))) | |
2 | frege67a 42245 | . 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 205 ∧ wa 397 if-wif 1062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 42150 ax-frege2 42151 ax-frege8 42169 ax-frege52a 42217 ax-frege58a 42235 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-tru 1545 df-fal 1555 |
This theorem is referenced by: (None) |
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