Theorem eqtrb

Description: A transposition of equality. (Contributed by Thierry Arnoux, 20-Aug-2023)

	Ref	Expression
Assertion	eqtrb	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) ↔ ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) → 𝐴 = 𝐵 )
2		eqtr2	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) → 𝐵 = 𝐶 )
3	1 2	jca	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) → ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) )
4		simpl	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) → 𝐴 = 𝐵 )
5		eqtr	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) → 𝐴 = 𝐶 )
6	4 5	jca	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) → ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) )
7	3 6	impbii	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) ↔ ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) )