Theorem elpwgded 41722

Description: elpwgdedVD 42075 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 4491	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	3	biimpar 481	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵)
5	1, 2, 4	syl2an 599	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2114  Vcvv 3398   ⊆ wss 3843  𝒫 cpw 4488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-in 3850  df-ss 3860  df-pw 4490

This theorem is referenced by:  sspwimp  42076  sspwimpALT  42083