Theorem infeq3

Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020)

	Ref	Expression
Assertion	infeq3	⊢ ( 𝑅 = 𝑆 → inf ( 𝐴 , 𝐵 , 𝑅 ) = inf ( 𝐴 , 𝐵 , 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		cnveq	⊢ ( 𝑅 = 𝑆 → ◡ 𝑅 = ◡ 𝑆 )
2		supeq3	⊢ ( ◡ 𝑅 = ◡ 𝑆 → sup ( 𝐴 , 𝐵 , ◡ 𝑅 ) = sup ( 𝐴 , 𝐵 , ◡ 𝑆 ) )
3	1 2	syl	⊢ ( 𝑅 = 𝑆 → sup ( 𝐴 , 𝐵 , ◡ 𝑅 ) = sup ( 𝐴 , 𝐵 , ◡ 𝑆 ) )
4		df-inf	⊢ inf ( 𝐴 , 𝐵 , 𝑅 ) = sup ( 𝐴 , 𝐵 , ◡ 𝑅 )
5		df-inf	⊢ inf ( 𝐴 , 𝐵 , 𝑆 ) = sup ( 𝐴 , 𝐵 , ◡ 𝑆 )
6	3 4 5	3eqtr4g	⊢ ( 𝑅 = 𝑆 → inf ( 𝐴 , 𝐵 , 𝑅 ) = inf ( 𝐴 , 𝐵 , 𝑆 ) )