Theorem elpwgded 42184

Description: elpwgdedVD 42537 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 4536	. . 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 3432   ⊆ wss 3887  𝒫 cpw 4533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535

This theorem is referenced by:  sspwimp  42538  sspwimpALT  42545