Modus Ponendo Ponens/Also known as

Proof Rule

Modus Ponens, abbreviated M.P.
The Rule of Implies-Elimination
The Rule of Arrow-Elimination
The Rule of Material Detachment, or just Rule of Detachment
The Process of Inference

1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$, seemingly uneasy with the language they are using, state:

The process of the inference cannot be reduced to symbols.

Having said that, they then go on to write:

... we shall write instead

"$\vdash p \supset \, \vdash q$,"

which is to be considered as a mere abbreviation of the threefold statement

"$\vdash p$" and "$\vdash \paren {p \supset q}$" and "$\vdash q$."

Remember that 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica use $\supset$ to denote the implication function.

Sources

1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$ ... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.1$: Formal Proof of Validity: Rules of Inference: $1.$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): logic
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): modus ponens
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logic
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): modus ponens