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Theorem elab3 2716
 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 2715 . 2 ((𝜓𝐴 ∈ V) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
41, 3ax-mp 7 1 (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   = wceq 1259   ∈ wcel 1409  {cab 2042  Vcvv 2574 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576 This theorem is referenced by:  elrnmpt2  5641  isfi  6271  addnqprlemfl  6714  addnqprlemfu  6715  mulnqprlemfl  6730  mulnqprlemfu  6731
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