Theorem infeq123d

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

	Ref	Expression
Hypotheses	infeq123d.a	⊢ ( 𝜑 → 𝐴 = 𝐷 )
	infeq123d.b	⊢ ( 𝜑 → 𝐵 = 𝐸 )
	infeq123d.c	⊢ ( 𝜑 → 𝐶 = 𝐹 )
Assertion	infeq123d	⊢ ( 𝜑 → inf ( 𝐴 , 𝐵 , 𝐶 ) = inf ( 𝐷 , 𝐸 , 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		infeq123d.a	⊢ ( 𝜑 → 𝐴 = 𝐷 )
2		infeq123d.b	⊢ ( 𝜑 → 𝐵 = 𝐸 )
3		infeq123d.c	⊢ ( 𝜑 → 𝐶 = 𝐹 )
4	3	cnveqd	⊢ ( 𝜑 → ◡ 𝐶 = ◡ 𝐹 )
5	1 2 4	supeq123d	⊢ ( 𝜑 → sup ( 𝐴 , 𝐵 , ◡ 𝐶 ) = sup ( 𝐷 , 𝐸 , ◡ 𝐹 ) )
6		df-inf	⊢ inf ( 𝐴 , 𝐵 , 𝐶 ) = sup ( 𝐴 , 𝐵 , ◡ 𝐶 )
7		df-inf	⊢ inf ( 𝐷 , 𝐸 , 𝐹 ) = sup ( 𝐷 , 𝐸 , ◡ 𝐹 )
8	5 6 7	3eqtr4g	⊢ ( 𝜑 → inf ( 𝐴 , 𝐵 , 𝐶 ) = inf ( 𝐷 , 𝐸 , 𝐹 ) )