Theorem elpwgded

Description: elpwgdedVD in conventional notation. (Contributed by Alan Sare, 23-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	elpwgded.1	$$⊢\phi \to A\in \mathrm{V}$$
	elpwgded.2	$$⊢\psi \to A\subseteq B$$
Assertion	elpwgded	$$⊢\left(\phi \wedge \psi \right)\to A\in 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		elpwgded.1	$$⊢\phi \to A\in \mathrm{V}$$
2		elpwgded.2	$$⊢\psi \to A\subseteq B$$
3		elpwg	$$⊢A\in \mathrm{V}\to \left(A\in 𝒫B\leftrightarrow A\subseteq B\right)$$
4	3	biimpar	$$⊢\left(A\in \mathrm{V}\wedge A\subseteq B\right)\to A\in 𝒫B$$
5	1 2 4	syl2an	$$⊢\left(\phi \wedge \psi \right)\to A\in 𝒫B$$