Theorem frege114d 44525

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 44744. (Contributed by RP, 15-Jul-2020.)

Hypothesis

Ref	Expression
frege114d.ab	⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))

Assertion

Ref	Expression
frege114d	⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴))

Proof of Theorem frege114d

Step	Hyp	Ref	Expression
1		frege114d.ab	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))
2		df-3or 1104	. . . 4 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ ((𝐴𝑅𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵𝑅𝐴))
3	2	biimpri 231	. . 3 ⊢ (((𝐴𝑅𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵𝑅𝐴) → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴))
4	3	orcs 889	. 2 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴 = 𝐵) → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴))
5	1, 4	syl 18	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   ∨ w3o 1102   = wceq 1570   class class class wbr 5114

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862  df-3or 1104

This theorem is used by:  frege111d  44526  frege126d  44529