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Theorem elpwgded 40775
Description: elpwgdedVD 41128 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 4541 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
43biimpar 478 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) → 𝐴 ∈ 𝒫 𝐵)
51, 2, 4syl2an 595 1 ((𝜑𝜓) → 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  Vcvv 3492  wss 3933  𝒫 cpw 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-in 3940  df-ss 3949  df-pw 4537
This theorem is referenced by:  sspwimp  41129  sspwimpALT  41136
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