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Theorem elabf2 11111
Description: One implication of elabf 2750. (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 2225 . . 3 𝑥𝐴
3 elabf2.nf . . 3 𝑥𝜓
4 elabf2.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜑))
52, 3, 4elabgf2 11109 . 2 (𝐴 ∈ V → (𝜓𝐴 ∈ {𝑥𝜑}))
61, 5ax-mp 7 1 (𝜓𝐴 ∈ {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wnf 1392  wcel 1436  {cab 2071  Vcvv 2615
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617
This theorem is referenced by:  elab2a  11113  bj-bdfindis  11271
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