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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege64a | Structured version Visualization version GIF version |
Description: Lemma for frege65a 41451. 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 41448 | . 2 ⊢ (if-(𝜎, 𝜒, 𝜂) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → if-(𝜎, 𝜃, 𝜁))) | |
2 | frege18 41386 | . 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 396 if-wif 1060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 41358 ax-frege2 41359 ax-frege8 41377 ax-frege58a 41443 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 |
This theorem is referenced by: frege65a 41451 |
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