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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 → 𝐴 ∈ V )
	elpwgded.2	⊢ ( 𝜓 → 𝐴 ⊆ 𝐵 )
Assertion	elpwgded	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ∈ 𝒫 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elpwgded.1	⊢ ( 𝜑 → 𝐴 ∈ V )
2		elpwgded.2	⊢ ( 𝜓 → 𝐴 ⊆ 𝐵 )
3		elpwg	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )
4	3	biimpar	⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ∈ 𝒫 𝐵 )
5	1 2 4	syl2an	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ∈ 𝒫 𝐵 )