In act 1, egeus begged for the ancient law of athens, which allowed him to choose hermia’s husband or have her executed. in act 4, this conflict is resolved. how?

The conflict is resolved because the four Athenians (Helena, Hermia, Lysander, and Demetrius) work themselves out (with the help of Puck and Oberon's magic) to be in couples that weren't in conflict with each other. When the four are discovered by Theseus and Egeus, Lysander says he doesn't want to marry Hermia anymore, he wants to marry Helena. Theseus agrees that they will get married alongside himself and Hippolyta.