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Theorem elpwgded 43315
Description: elpwgdedVD 43668 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpwgded.1 (𝜑𝐴 ∈ V)
elpwgded.2 (𝜓𝐴𝐵)
Assertion
Ref Expression
elpwgded ((𝜑𝜓) → 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwgded
StepHypRef Expression
1 elpwgded.1 . 2 (𝜑𝐴 ∈ V)
2 elpwgded.2 . 2 (𝜓𝐴𝐵)
3 elpwg 4605 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
43biimpar 478 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) → 𝐴 ∈ 𝒫 𝐵)
51, 2, 4syl2an 596 1 ((𝜑𝜓) → 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3474  wss 3948  𝒫 cpw 4602
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 3955  df-ss 3965  df-pw 4604
This theorem is referenced by:  sspwimp  43669  sspwimpALT  43676
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