Theorem elpwgded 43310

Description: elpwgdedVD 43663 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elpwgded.1	⊢ (𝜑 → 𝐴 ∈ V)
elpwgded.2	⊢ (𝜓 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
elpwgded	⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwgded

Step	Hyp	Ref	Expression
1		elpwgded.1	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
2		elpwgded.2	. 2 ⊢ (𝜓 → 𝐴 ⊆ 𝐵)
3		elpwg 4604	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	3	biimpar 478	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵)
5	1, 2, 4	syl2an 596	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3947  𝒫 cpw 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-pw 4603

This theorem is referenced by:  sspwimp  43664  sspwimpALT  43671