Theorem frege98d

Description: If C follows A and B follows C in the transitive closure of R , then B follows A in the transitive closure of R . Similar to Proposition 98 of Frege1879 p. 71. Compare with frege98 . (Contributed by RP, 15-Jul-2020)

	Ref	Expression
Hypotheses	frege98d.a	⊢ ( 𝜑 → 𝐴 ∈ V )
	frege98d.b	⊢ ( 𝜑 → 𝐵 ∈ V )
	frege98d.c	⊢ ( 𝜑 → 𝐶 ∈ V )
	frege98d.ac	⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝑅 ) 𝐶 )
	frege98d.cb	⊢ ( 𝜑 → 𝐶 ( t+ ‘ 𝑅 ) 𝐵 )
Assertion	frege98d	⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝑅 ) 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		frege98d.a	⊢ ( 𝜑 → 𝐴 ∈ V )
2		frege98d.b	⊢ ( 𝜑 → 𝐵 ∈ V )
3		frege98d.c	⊢ ( 𝜑 → 𝐶 ∈ V )
4		frege98d.ac	⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝑅 ) 𝐶 )
5		frege98d.cb	⊢ ( 𝜑 → 𝐶 ( t+ ‘ 𝑅 ) 𝐵 )
6		brcogw	⊢ ( ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V ) ∧ ( 𝐴 ( t+ ‘ 𝑅 ) 𝐶 ∧ 𝐶 ( t+ ‘ 𝑅 ) 𝐵 ) ) → 𝐴 ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) 𝐵 )
7	1 2 3 4 5 6	syl32anc	⊢ ( 𝜑 → 𝐴 ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) 𝐵 )
8		trclfvcotrg	⊢ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 )
9	8	a1i	⊢ ( 𝜑 → ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) )
10	9	ssbrd	⊢ ( 𝜑 → ( 𝐴 ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) 𝐵 → 𝐴 ( t+ ‘ 𝑅 ) 𝐵 ) )
11	7 10	mpd	⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝑅 ) 𝐵 )