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Theorem elab3 2768
 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 2767 . 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 104   = wceq 1290   ∈ wcel 1439  {cab 2075  Vcvv 2620 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622 This theorem is referenced by:  elrnmpt2  5772  isfi  6532  addnqprlemfl  7179  addnqprlemfu  7180  mulnqprlemfl  7195  mulnqprlemfu  7196  istps  11791
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