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Theorem elab 2828
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
Hypotheses
Ref Expression
elab.1 𝐴 ∈ V
elab.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elab (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elab
StepHypRef Expression
1 nfv 1508 . 2 𝑥𝜓
2 elab.1 . 2 𝐴 ∈ V
3 elab.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elabf 2827 1 (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  {cab 2125  Vcvv 2686
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 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  ralab  2844  rexab  2846  intab  3800  dfiin2g  3846  dfiunv2  3849  uniuni  4372  dcextest  4495  peano5  4512  finds  4514  finds2  4515  funcnvuni  5192  fun11iun  5388  elabrex  5659  abrexco  5660  mapval2  6572  ssenen  6745  snexxph  6838  sbthlem2  6846  indpi  7162  nqprm  7362  nqprrnd  7363  nqprdisj  7364  nqprloc  7365  nqprl  7371  nqpru  7372  cauappcvgprlem2  7480  caucvgprlem2  7500  peano1nnnn  7672  peano2nnnn  7673  1nn  8743  peano2nn  8744  dfuzi  9173  hashfacen  10591  shftfvalg  10602  ovshftex  10603  shftfval  10605  txdis1cn  12461  bj-ssom  13218
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