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Metamath Proof Explorer

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	$⊢ φ \to A \in V$
	elpwgded.2	$⊢ ψ \to A \subseteq B$
Assertion	elpwgded	$⊢ (φ \land ψ) \to A \in 𝒫 B$

Proof

Step	Hyp	Ref	Expression
1		elpwgded.1	$⊢ φ \to A \in V$
2		elpwgded.2	$⊢ ψ \to A \subseteq B$
3		elpwg	$⊢ A \in V \to (A \in 𝒫 B \leftrightarrow A \subseteq B)$
4	3	biimpar	$⊢ (A \in V \land A \subseteq B) \to A \in 𝒫 B$
5	1 2 4	syl2an	$⊢ (φ \land ψ) \to A \in 𝒫 B$