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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege114d | Structured version Visualization version GIF version |
Description: If either 𝑅 relates 𝐴 and 𝐵 or 𝐴 and 𝐵 are the same, then either 𝐴 and 𝐵 are the same, 𝑅 relates 𝐴 and 𝐵, 𝑅 relates 𝐵 and 𝐴. Similar to Proposition 114 of [Frege1879] p. 76. Compare with frege114 40321. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege114d.ab | ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) |
Ref | Expression |
---|---|
frege114d | ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege114d.ab | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) | |
2 | df-3or 1084 | . . . 4 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ ((𝐴𝑅𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵𝑅𝐴)) | |
3 | 2 | biimpri 230 | . . 3 ⊢ (((𝐴𝑅𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵𝑅𝐴) → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)) |
4 | 3 | orcs 871 | . 2 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴 = 𝐵) → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∨ w3o 1082 = wceq 1533 class class class wbr 5065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 209 df-or 844 df-3or 1084 |
This theorem is referenced by: frege111d 40102 frege126d 40105 |
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