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Theorem elpwgded 42938
Description: elpwgdedVD 43291 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 4567 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
43biimpar 479 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) → 𝐴 ∈ 𝒫 𝐵)
51, 2, 4syl2an 597 1 ((𝜑𝜓) → 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Vcvv 3447  wss 3914  𝒫 cpw 4564
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566
This theorem is referenced by:  sspwimp  43292  sspwimpALT  43299
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