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Theorem elintrabg 3675
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
Assertion
Ref Expression
elintrabg (𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem elintrabg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2145 . 2 (𝑦 = 𝐴 → (𝑦 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥𝐵𝜑}))
2 eleq1 2145 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 228 . . 3 (𝑦 = 𝐴 → ((𝜑𝑦𝑥) ↔ (𝜑𝐴𝑥)))
43ralbidv 2374 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 (𝜑𝑦𝑥) ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
5 vex 2615 . . 3 𝑦 ∈ V
65elintrab 3674 . 2 (𝑦 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝑦𝑥))
71, 4, 6vtoclbg 2670 1 (𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  wral 2353  {crab 2357   cint 3662
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-ral 2358  df-rab 2362  df-v 2614  df-int 3663
This theorem is referenced by: (None)
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