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Theorem elintrabg 3628
 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 2100 . 2 (𝑦 = 𝐴 → (𝑦 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥𝐵𝜑}))
2 eleq1 2100 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 219 . . 3 (𝑦 = 𝐴 → ((𝜑𝑦𝑥) ↔ (𝜑𝐴𝑥)))
43ralbidv 2326 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 (𝜑𝑦𝑥) ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
5 vex 2560 . . 3 𝑦 ∈ V
65elintrab 3627 . 2 (𝑦 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝑦𝑥))
71, 4, 6vtoclbg 2614 1 (𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310  ∩ cint 3615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-int 3616 This theorem is referenced by: (None)
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