Theorem elpwgded 44535

Description: elpwgdedVD 44888 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 4625	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	3	biimpar 477	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵)
5	1, 2, 4	syl2an 595	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ss 3993  df-pw 4624

This theorem is referenced by:  sspwimp  44889  sspwimpALT  44896