Theorem 19.32v

Version of 19.32 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)

Conclusion
\( \vdash \forall x \, (\varphi \vee \psi) \leftrightarrow \varphi \vee \forall x \, \psi \)
Proof
l0
19.21v
\( \vdash \forall x \, (\neg \varphi \rightarrow \psi) \leftrightarrow \neg \varphi \rightarrow \forall x \, \psi \)
df-or
l2
albii _
\( \vdash \forall x \, (\varphi \vee \psi) \leftrightarrow \forall x \, (\neg \varphi \rightarrow \psi) \)
df-or
\( \vdash \varphi \vee \forall x \, \psi \leftrightarrow \neg \varphi \rightarrow \forall x \, \psi \)
3bitr4i l0 l2 _