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Theorem elab4g 2898
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
Hypotheses
Ref Expression
elab4g.1 (𝑥 = 𝐴 → (𝜑𝜓))
elab4g.2 𝐵 = {𝑥𝜑}
Assertion
Ref Expression
elab4g (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elab4g
StepHypRef Expression
1 elex 2760 . 2 (𝐴𝐵𝐴 ∈ V)
2 elab4g.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
3 elab4g.2 . . 3 𝐵 = {𝑥𝜑}
42, 3elab2g 2896 . 2 (𝐴 ∈ V → (𝐴𝐵𝜓))
51, 4biadan2 456 1 (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  {cab 2173  Vcvv 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751
This theorem is referenced by: (None)
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