query: Clarifying on how if p,q is logically equivalent to p only if q
positive_passages.docid: 29031
positive_passages.text: How does "If $P$ then $Q$" have the same meaning as "$Q$ only if $P$ "?
negative_passages.docid: 62936
negative_passages.text: Proving equivalencies of two versions of the Axiom of Choice
negative_passages.docid: 42150
negative_passages.text: Prove that $\forall x \neg P(x) \iff \neg \exists x P(x)$
negative_passages.docid: 124742
negative_passages.text: P(A|C)=P(A|B)*P(B|C)?
negative_passages.docid: 46939
negative_passages.text: Natural deduction, Proof $\vdash$ $P\Rightarrow(Q\Rightarrow P)$
negative_passages.docid: 45323
negative_passages.text: How to prove $\vdash\neg P\to (P\to Q)$?
negative_passages.docid: 65379
negative_passages.text: Deriving $\neg R$ from $\{R↔(R∨(P∧¬P)), R↔¬P, ¬P→(P↔(Q→Q)), P→Q\}$
negative_passages.docid: 41846
negative_passages.text: $(\mathbb{R}^n,\|\cdot\|_{p})$ is isometrically isomorphic to $(\mathbb{R}^n,\|\cdot\|_{q})$ iff $p=q,$
negative_passages.docid: 87955
negative_passages.text: How to prove that $P \rightarrow Q$ is equivalent with $\neg P \lor Q $?
negative_passages.docid: 150053
negative_passages.text: Is "$(P \land Q) \to P$" a tautology?
negative_passages.docid: 32009
negative_passages.text: Prove $\;\big((p\rightarrow q) \lor (p \rightarrow r)\big) \rightarrow (q\lor r)\equiv p \lor q \lor r$ without use of a truth table.
negative_passages.docid: 139064
negative_passages.text: Show by using logical connectives laws that $(P\to Q) \land (Q \to R) $ is equivalent to $(P \to R) \land [(P \iff Q) \lor (R \iff Q)]$
negative_passages.docid: 37183
negative_passages.text: Show that the set of all $x$ such that $x \in P$ and $x \notin Q$ exists
negative_passages.docid: 144225
negative_passages.text: $\ell_p$ is Hilbert if and only if $p=2$
negative_passages.docid: 25187
negative_passages.text: Why $∀x(P(x)→Q)$ is equivalent to $∃xP(x)→Q$
negative_passages.docid: 48614
negative_passages.text: Why is "$P \Rightarrow Q$" equivalent to "$\neg Q \Rightarrow \neg P$"?
negative_passages.docid: 157625
negative_passages.text: How to prove conditional in Coq?