Let {X-i (t), t >= 0}, i = 1, 2 be two standard fractional Brownian motions being jointly Gaussian with constant cross-correlation. In this paper, we derive the exact asymptotics of the joint survival function
P {sups(is an element of)[(0,1]) X-1(s) > u, sup(t is an element of)[(0,1]) X-2(t) > u}
as u ->infinity. A novel finding of this contribution is the exponential approximation of the joint conditional first passage times of X-1, X-2. As a by-product, we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields. Keywords Borell-TIS inequality; Extremes; First passage times; Fractional Brownian motion; Gaussian random fields; Piterbarg inequality.