Theorem 19.41v

Version of 19.41 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) Remove dependency on [ax-6]. (Revised by Rohan Ridenour, 15-Apr-2022.)

Conclusion
\( \vdash \exists x \, (\varphi \wedge \psi) \leftrightarrow \exists x \, \varphi \wedge \psi \)
Proof
l0
19.40
\( \vdash \exists x \, (\varphi \wedge \psi) \rightarrow \exists x \, \varphi \wedge \exists x \, \psi \)
ax5e
anim2i _
\( \vdash \exists x \, \varphi \wedge \exists x \, \psi \rightarrow \exists x \, \varphi \wedge \psi \)
l3
syl l0 _
pm3.21
eximdv _
impcom _
impbii l3 _