show that 7(p->q) and p^7 q are logically equivalent without using truth table or using identities

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We want to start with one of the statements, and transform it into the other through a sequence of logically equivalent statements. Start with ¬(P→Q)

We can rewrite the implication as a disjunction this is logically equivalent to

¬(¬P∨Q)

Now apply DeMorgan's law to get

¬¬P∧¬Q.

Finally, use double negation to arrive at P∧¬Q