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Theorem elpwgdedVD 39671
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4311. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 39301 is elpwgdedVD 39671 using conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpwgdedVD.1 (   𝜑   ▶   𝐴 ∈ V   )
elpwgdedVD.2 (   𝜓   ▶   𝐴𝐵   )
Assertion
Ref Expression
elpwgdedVD (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )

Proof of Theorem elpwgdedVD
StepHypRef Expression
1 elpwgdedVD.1 . 2 (   𝜑   ▶   𝐴 ∈ V   )
2 elpwgdedVD.2 . 2 (   𝜓   ▶   𝐴𝐵   )
3 elpwg 4311 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
43biimpar 503 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) → 𝐴 ∈ 𝒫 𝐵)
51, 2, 4el12 39474 1 (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )
Colors of variables: wff setvar class
Syntax hints:  wcel 2140  Vcvv 3341  wss 3716  𝒫 cpw 4303  (   wvd1 39306  (   wvhc2 39317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-in 3723  df-ss 3730  df-pw 4305  df-vd1 39307  df-vhc2 39318
This theorem is referenced by:  sspwimpVD  39673
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