Theorem elpwgded 39539

Description: elpwgdedVD 39902 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 4356	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	3	biimpar 470	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵)
5	1, 2, 4	syl2an 590	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 385   ∈ wcel 2157  Vcvv 3384   ⊆ wss 3768  𝒫 cpw 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2776

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-v 3386  df-in 3775  df-ss 3782  df-pw 4350

This theorem is referenced by:  sspwimp  39903  sspwimpALT  39910