Integral equation for Probability that A realizes before B AND B realizes before C $A$: random variable to denote time to failure for equipment E_A
$B$: random variable to denote time to failure for equipment E_B
$A$ and $B$ are independent.
$P(A◁B)(T)$ : Probability that A realizes before B in time range (0, T).
I know & understand the equation below (from a paper):
$$P(A◁B)(T)=\int_0^T \biggl( pdf_A(t)*(1-CDF_B(t)) \biggr) dt$$
I couldn't figure the integral structure for the case that there exists 3rd random variable:
$C$: random variable to denote time to failure for equipment E_C
$A$ and $B$ and $C$ are independent.
$P(A◁B◁C)(T)$ : Probability that A realizes before B AND B realizes before C in time range (0, T).
$$P(A◁B◁C)(T)=INTEGRAL\ STRUCTURE$$
What is the INTEGRAL STRUCTURE in this 3 variable case?
Do i need a double integral something or does my trial below correct?
$$P(A◁B◁C)(T)=\int_0^T \biggl( pdf_A(t)*(1-CDF_B(t)) \biggr) dt \times \int_0^T \biggl( pdf_B(t)*(1-CDF_C(t)) \biggr) dt$$
thanks. regards.

Your question is ambiguous about which variables occur prior to time $T$ in the three variable case.  Based on your comment below, the event of interest is that all three events occur by time $T$ and they occur in the stated order:
$$\{A ◁ B ◁ C\}(T) \quad \quad \iff \quad \quad 
A < B < C \leqslant T.$$
Assuming that $A,B,C$ are mutually independent non-negative random variables, and using your notation for the events of interest (but standard notation for the PDFs and CDFs) you would have:
$$\mathbb{P}(A ◁ B ◁ C)(T) = \int \limits_0^T \Bigg( \int \limits_a^T p_B(b) (F_C(T)-F_C(b)) \ db \Bigg) p_A(a) \ da.$$
