Theorem mpan

An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses
mpan.1
\( \vdash \varphi \)
mpan.2
\( \vdash \varphi \wedge \psi \rightarrow \chi \)
Conclusion
\( \vdash \psi \rightarrow \chi \)
Proof
mpan.1
l1
a1i _
\( \vdash \psi \rightarrow \varphi \)
mpan.2
\( \vdash \varphi \wedge \psi \rightarrow \chi \)
mpancom l1 _