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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege66a | Structured version Visualization version GIF version |
Description: Swap antecedents of frege65a 37197. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege66a | ⊢ (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege65a 37197 | . 2 ⊢ (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) | |
2 | ax-frege8 37123 | . 2 ⊢ ((((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 if-wif 1006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 37104 ax-frege2 37105 ax-frege8 37123 ax-frege58a 37189 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 |
This theorem is referenced by: (None) |
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