Theorem ralsn

Description: Convert a universal quantification restricted to a singleton to a substitution. (Contributed by NM, 27-Apr-2009)

	Ref	Expression
Hypotheses	ralsn.1	$$⊢A\in \mathrm{V}$$
	ralsn.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	ralsn	$$⊢\forall x\in \left\{A\right\}\phi \leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		ralsn.1	$$⊢A\in \mathrm{V}$$
2		ralsn.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3	2	ralsng	$$⊢A\in \mathrm{V}\to \left(\forall x\in \left\{A\right\}\phi \leftrightarrow \psi \right)$$
4	1 3	ax-mp	$$⊢\forall x\in \left\{A\right\}\phi \leftrightarrow \psi$$