Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  elabf2 GIF version

Theorem elabf2 11024
Description: One implication of elabf 2747. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabf2.nf 𝑥𝜓
elabf2.s 𝐴 ∈ V
elabf2.1 (𝑥 = 𝐴 → (𝜓𝜑))
Assertion
Ref Expression
elabf2 (𝜓𝐴 ∈ {𝑥𝜑})
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem elabf2
StepHypRef Expression
1 elabf2.s . 2 𝐴 ∈ V
2 nfcv 2223 . . 3 𝑥𝐴
3 elabf2.nf . . 3 𝑥𝜓
4 elabf2.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜑))
52, 3, 4elabgf2 11022 . 2 (𝐴 ∈ V → (𝜓𝐴 ∈ {𝑥𝜑}))
61, 5ax-mp 7 1 (𝜓𝐴 ∈ {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wnf 1390  wcel 1434  {cab 2069  Vcvv 2612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614
This theorem is referenced by:  elab2a  11026  bj-bdfindis  11184
  Copyright terms: Public domain W3C validator