Consider the statement
(P∧Q≡P)⇔(Q≡⊤)
Where P
and Q
are statements, and ⊤
denotes the tautology (true) statement. It seems intuitively true that the above biconditional statement is true. But I would like to prove it.
One direction is easy enough to prove: suppose Q≡⊤
; then by substitution, we have
P∧Q ≡ P∧⊤ ≡ P
by the identity law for conjunction.
However, the converse seems trickier. By assuming P∧Q≡P
, how would one know for certain that Q≡⊤
?
Thanks in advance for your thoughts.
(P.S. Is it improper of me to use two different symbols for equivalence here?)