I took a deductive logic course
in the 70s. I think the notation we used was different than yours, since I can't make heads or tails of your demonstration. (other possibility is that I've forgotten everything I learned)
About all I recognized was your if-then. Can you explain your use of parenthesis, the lower case 'i', the carat (nagging suspicion I should recognize the carat), the upper-case 'X'?