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

Theorem elabOLD 3572
 Description: Obsolete version of elab 3571 as of 29-Nov-2022. Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elab.1 𝐴 ∈ V
elab.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elabOLD (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elabOLD
StepHypRef Expression
1 nfv 2015 . 2 𝑥𝜓
2 elab.1 . 2 𝐴 ∈ V
3 elab.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elabf 3568 1 (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1658   ∈ wcel 2166  {cab 2811  Vcvv 3414 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator