Theorem exbidh

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

	Ref	Expression
Hypotheses	exbidh.1	$$⊢\phi \to \forall x\phi$$
	exbidh.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	exbidh	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists x\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		exbidh.1	$$⊢\phi \to \forall x\phi$$
2		exbidh.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	2	alexbii	$$⊢\forall x\phi \to \left(\exists x\psi \leftrightarrow \exists x\chi \right)$$
4	1 3	syl	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists x\chi \right)$$