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Theorem elab4g 3668
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 3487 . 2 (𝐴𝐵𝐴 ∈ V)
2 elab4g.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
3 elab4g.2 . . 3 𝐵 = {𝑥𝜑}
42, 3elab2g 3665 . 2 (𝐴 ∈ V → (𝐴𝐵𝜓))
51, 4biadanii 819 1 (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  {cab 2703  Vcvv 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470
This theorem is referenced by:  isprs  18260  ispos  18277  istrkgc  28209  istrkgb  28210  istrkgcb  28211  istrkge  28212  istrkgl  28213  eulerpartlemt0  33898  istrkg2d  34207
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