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Theorem elab3 2831
 Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
Hypotheses
Ref Expression
elab3.1 (𝜓𝐴 ∈ V)
elab3.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elab3 (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elab3
StepHypRef Expression
1 elab3.1 . 2 (𝜓𝐴 ∈ V)
2 elab3.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32elab3g 2830 . 2 ((𝜓𝐴 ∈ V) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
41, 3ax-mp 5 1 (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2123  Vcvv 2681 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683 This theorem is referenced by:  elrnmpo  5877  isfi  6648  addnqprlemfl  7360  addnqprlemfu  7361  mulnqprlemfl  7376  mulnqprlemfu  7377  istps  12188
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