Theorem exbidh

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

Hypotheses
exbidh.1
\( \vdash \varphi \rightarrow \forall x \, \varphi \)
exbidh.2
\( \vdash \varphi \rightarrow (\psi \leftrightarrow \chi) \)
Conclusion
\( \vdash \varphi \rightarrow (\exists x \, \psi \leftrightarrow \exists x \, \chi) \)
Proof
l0
exbidh.1
\( \vdash \varphi \rightarrow \forall x \, \varphi \)
exbidh.2
alexbii _
\( \vdash \forall x \, \varphi \rightarrow (\exists x \, \psi \leftrightarrow \exists x \, \chi) \)
syl l0 _