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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege60a | Structured version Visualization version GIF version |
Description: Swap antecedents of ax-frege58a 43199. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege60a | ⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege58acor 43200 | . . 3 ⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, (𝜒 → 𝜃), (𝜂 → 𝜁)))) | |
2 | ifpimim 42833 | . . 3 ⊢ (if-(𝜑, (𝜒 → 𝜃), (𝜂 → 𝜁)) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁))) | |
3 | 1, 2 | syl6 35 | . 2 ⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜓, 𝜏) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁)))) |
4 | frege12 43137 | . 2 ⊢ ((((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜓, 𝜏) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁)))) → (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 if-wif 1059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 43114 ax-frege2 43115 ax-frege8 43133 ax-frege58a 43199 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 |
This theorem is referenced by: (None) |
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