![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > frege64a | Structured version Visualization version GIF version |
Description: Lemma for frege65a 42243. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege64a | ⊢ ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege62a 42240 | . 2 ⊢ (if-(𝜎, 𝜒, 𝜂) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → if-(𝜎, 𝜃, 𝜁))) | |
2 | frege18 42178 | . 2 ⊢ ((if-(𝜎, 𝜒, 𝜂) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → if-(𝜎, 𝜃, 𝜁))) → ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁))))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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-frege58a 42235 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 |
This theorem is referenced by: frege65a 42243 |
Copyright terms: Public domain | W3C validator |