I am trying to prove that $[a\Rightarrow(b\lor c)]\Leftrightarrow[(a\land\lnot b)\Rightarrow c]$.
My proof is the following:
- $a\Rightarrow(b\lor c)~$ Premise
- $(a\Rightarrow b)\lor c~$ Associative Law
- $(\lnot a\lor b)\lor c~$ Material Implication
- $\lnot(a\lor\lnot b)\lor c~$ De Morgan's Law
- $(a\land\lnot b)\Rightarrow c~$ Material Implication
I'm having doubts about my second step. I tried to check for the validity of my step using truth table, and the statements in the first and second steps are logically equivalent. Is my application of associative law legal?