# Material conditional

THEOREM    $$P \Rightarrow (Q \Rightarrow R) \vdash (P \Rightarrow Q) \Rightarrow (P \Rightarrow R)$$

Proof available

THEOREM    $$(P \Rightarrow Q) \Rightarrow (P \Rightarrow R) \vdash P \Rightarrow (Q \Rightarrow R)$$

Proof available

THEOREM

The material conditional operation is left self-distributive.

$$P \Rightarrow (Q \Rightarrow R) \dashv\vdash (P \Rightarrow Q) \Rightarrow (P \Rightarrow R)$$

Proof available

THEOREM

The material conditional relation is reflexive.

$$\vdash P \Rightarrow P$$

Proof available

THEOREM (Hypothetical syllogism)

The material conditional relation is transitive.

$$P \Rightarrow Q,\ Q \Rightarrow R \vdash P \Rightarrow R$$

Proof available

THEOREM

Material conditional is left-distributive over disjunction.

$$P \Rightarrow Q \vee R \dashv\vdash (P \Rightarrow Q) \vee (P \Rightarrow R)$$

Proof unavailable

THEOREM

Material conditional is right-distributive over conjunction.

$$P \Rightarrow Q \wedge R \dashv\vdash (P \Rightarrow Q) \wedge (P \Rightarrow R)$$

Proof unavailable

THEOREM    $$P \wedge Q \Rightarrow R \dashv\vdash (P \Rightarrow R) \vee (Q \Rightarrow R)$$

Proof available

THEOREM    $$P \vee Q \Rightarrow R \dashv\vdash (P \Rightarrow R) \wedge (Q \Rightarrow R)$$

Proof available