IC3 is undoubtedly one of the most successful and important recent techniques for unbounded model checking. Understanding and improving IC3 has been a subject of a lot of recent research. In this regard, the most fundamental questions are how to choose Counterexamples to Induction (CTIs) and how to generalize them into (blocking) lemmas. Answers to both questions influence performance of the algorithm by directly affecting the quality of the lemmas learned. In this paper, we present a new IC3-based algorithm, called QUIP1, that is designed to more aggressively propagate (or push) learned lemmas to obtain a safe inductive invariant faster. QUIP modifies the recursive blocking procedure of IC3 to prioritize pushing already discovered lemmas over learning of new ones. However, a naive implementation of this strategy floods the algorithm with too many useless lemmas. In QUIP, we solve this by extending IC3 with may-proof-obligations (corresponding to the negations of learned lemmas), and by using an under-Approximation of reachable states (i.e., states that witness why a may-proofobligation is satisfiable) to prune non-inductive lemmas. We have implemented QUIP on top of an industrial-strength implementation of IC3. The experimental evaluation on HWMCC benchmarks shows that the QUIP is a significant improvement (at least 2x in runtime and more properties solved) over IC3. Furthermore, the new reasoning capabilities of QUIP naturally lead to additional optimizations and new techniques that can lead to further improvements in the future.