Load instructions often limit instruction-level parallelism (ILP) in modern processors due to data and resource dependences they cause. Prior techniques like Load Value Prediction (LVP) and Memory Renaming (MRN) mitigate load data dependence by predicting the data value of a load instruction. However, they fail to mitigate load resource dependence as the predicted load instruction gets executed nonetheless (even on a correct prediction), which consumes hard-to-scale pipeline resources that otherwise could have been used to execute other load instructions. Our goal in this work is to improve ILP by mitigating both load data dependence and resource dependence. To this end, we propose a purely-microarchitectural technique called Constable, that safely eliminates the execution of load instructions. Constable dynamically identifies load instructions that have repeatedly fetched the same data from the same load address. We call such loads likely-stable. For every likely-stable load, Constable (1) tracks modifications to its source architectural registers and memory location via lightweight hardware structures, and (2) eliminates the execution of subsequent instances of the load instruction until there is a write to its source register or a store or snoop request to its load address. Our extensive evaluation using a wide variety of 90 workloads shows that Constable improves performance by $5.1 \%$ while reducing the core dynamic power consumption by $3.4 \%$ on average over a strong baseline system that implements MRN and other dynamic instruction optimizations (e.g., move and zero elimination, constant and branch folding). In presence of 2-way simultaneous multithreading (SMT), Constable’s performance improvement increases to $8.8 \%$ over the baseline system. When combined with a state-of-the-art load value predictor (EVES), Constable provides an additional $3.7 \%$ and $7.8 \%$ average performance benefit over the load value predictor alone, in the baseline system without and with 2-way SMT, respectively.