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Theorem elab4g 3683
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
Hypotheses
Ref Expression
elab4g.1 (𝑥 = 𝐴 → (𝜑𝜓))
elab4g.2 𝐵 = {𝑥𝜑}
Assertion
Ref Expression
elab4g (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elab4g
StepHypRef Expression
1 elex 3501 . 2 (𝐴𝐵𝐴 ∈ V)
2 elab4g.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
3 elab4g.2 . . 3 𝐵 = {𝑥𝜑}
42, 3elab2g 3680 . 2 (𝐴 ∈ V → (𝐴𝐵𝜓))
51, 4biadanii 822 1 (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482
This theorem is referenced by:  isprs  18342  ispos  18360  istrkgc  28462  istrkgb  28463  istrkgcb  28464  istrkge  28465  istrkgl  28466  eulerpartlemt0  34371  istrkg2d  34681
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