MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elab4g Structured version   Visualization version   GIF version

Theorem elab4g 3685
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 3498 . 2 (𝐴𝐵𝐴 ∈ V)
2 elab4g.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
3 elab4g.2 . . 3 𝐵 = {𝑥𝜑}
42, 3elab2g 3682 . 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 1536  wcel 2105  {cab 2711  Vcvv 3477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479
This theorem is referenced by:  isprs  18353  ispos  18371  istrkgc  28476  istrkgb  28477  istrkgcb  28478  istrkge  28479  istrkgl  28480  eulerpartlemt0  34350  istrkg2d  34659
  Copyright terms: Public domain W3C validator