Theorem mprgbir

Description: Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004)

	Ref	Expression
Hypotheses	mprgbir.1	$$⊢\phi \leftrightarrow \forall x\in A\psi$$
	mprgbir.2	$$⊢x\in A\to \psi$$
Assertion	mprgbir	$$⊢\phi$$

Proof

Step	Hyp	Ref	Expression
1		mprgbir.1	$$⊢\phi \leftrightarrow \forall x\in A\psi$$
2		mprgbir.2	$$⊢x\in A\to \psi$$
3	2	rgen	$$⊢\forall x\in A\psi$$
4	3 1	mpbir	$$⊢\phi$$