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Theorem elab4g 2879
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 2741 . 2 (𝐴𝐵𝐴 ∈ V)
2 elab4g.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
3 elab4g.2 . . 3 𝐵 = {𝑥𝜑}
42, 3elab2g 2877 . 2 (𝐴 ∈ V → (𝐴𝐵𝜓))
51, 4biadan2 453 1 (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  {cab 2156  Vcvv 2730
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732
This theorem is referenced by: (None)
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