When you're proving the minimax thm, it seems like you need to prove #2 and #3 before proving #1. You stated that the payoff for the NE must be at least as high as the payoff for the maxMin, but the NE is just the best response to the strategy that the other player is playing in the Nash equilibrium, right? So it's not necessarily a best response to player 2's minimization strategy (unless player 2's min strategy is equal to player 2's NE strategy). Therefore, you need to prove #3 first, which says that player 2's NE strategy is equivalent to the minimization strategy of player 2, right?
When you're proving the minimax thm, it seems like you need to prove #2 and #3 before proving #1. You stated that the payoff for the NE must be at least as high as the payoff for the maxMin, but the NE is just the best response to the strategy that the other player is playing in the Nash equilibrium, right? So it's not necessarily a best response to player 2's minimization strategy (unless player 2's min strategy is equal to player 2's NE strategy). Therefore, you need to prove #3 first, which says that player 2's NE strategy is equivalent to the minimization strategy of player 2, right?