Theorem infeq3

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

	Ref	Expression
Assertion	infeq3	$$⊢R=S\to inf\left(A,B,R\right)=inf\left(A,B,S\right)$$

Proof

Step	Hyp	Ref	Expression
1		cnveq	$$⊢R=S\to R^{-1}=S^{-1}$$
2		supeq3	$$⊢R^{-1}=S^{-1}\to sup\left(A,B,R^{-1}\right)=sup\left(A,B,S^{-1}\right)$$
3	1 2	syl	$$⊢R=S\to sup\left(A,B,R^{-1}\right)=sup\left(A,B,S^{-1}\right)$$
4		df-inf	$$⊢inf\left(A,B,R\right)=sup\left(A,B,R^{-1}\right)$$
5		df-inf	$$⊢inf\left(A,B,S\right)=sup\left(A,B,S^{-1}\right)$$
6	3 4 5	3eqtr4g	$$⊢R=S\to inf\left(A,B,R\right)=inf\left(A,B,S\right)$$