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Theorem frege114d 38367
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 38588. (Contributed by RP, 15-Jul-2020.)
Hypothesis
Ref Expression
frege114d.ab (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))
Assertion
Ref Expression
frege114d (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵𝐵𝑅𝐴))

Proof of Theorem frege114d
StepHypRef Expression
1 frege114d.ab . 2 (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))
2 df-3or 1055 . . . 4 ((𝐴𝑅𝐵𝐴 = 𝐵𝐵𝑅𝐴) ↔ ((𝐴𝑅𝐵𝐴 = 𝐵) ∨ 𝐵𝑅𝐴))
32biimpri 218 . . 3 (((𝐴𝑅𝐵𝐴 = 𝐵) ∨ 𝐵𝑅𝐴) → (𝐴𝑅𝐵𝐴 = 𝐵𝐵𝑅𝐴))
43orcs 408 . 2 ((𝐴𝑅𝐵𝐴 = 𝐵) → (𝐴𝑅𝐵𝐴 = 𝐵𝐵𝑅𝐴))
51, 4syl 17 1 (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  w3o 1053   = wceq 1523   class class class wbr 4685
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 197  df-or 384  df-3or 1055
This theorem is referenced by:  frege111d  38368  frege126d  38371
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