Theorem albidh

Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 26-May-1993.)

Hypotheses
albidh.1
\( \vdash \varphi \rightarrow \forall x \, \varphi \)
albidh.2
\( \vdash \varphi \rightarrow (\psi \leftrightarrow \chi) \)
Conclusion
\( \vdash \varphi \rightarrow (\forall x \, \psi \leftrightarrow \forall x \, \chi) \)
Proof
l0
albidh.1
albidh.2
l2
alrimih l0 _
\( \vdash \varphi \rightarrow \forall x \, (\psi \leftrightarrow \chi) \)
albi
\( \vdash \forall x \, (\psi \leftrightarrow \chi) \rightarrow (\forall x \, \psi \leftrightarrow \forall x \, \chi) \)
syl l2 _