Theorem albidh

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

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

Proof

Step	Hyp	Ref	Expression
1		albidh.1	$$⊢\phi \to \forall x\phi$$
2		albidh.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	1 2	alrimih	$$⊢\phi \to \forall x\left(\psi \leftrightarrow \chi \right)$$
4		albi	$$⊢\forall x\left(\psi \leftrightarrow \chi \right)\to \left(\forall x\psi \leftrightarrow \forall x\chi \right)$$
5	3 4	syl	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall x\chi \right)$$