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Theorem elintrabg 4424
 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 2676 . 2 (𝑦 = 𝐴 → (𝑦 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥𝐵𝜑}))
2 eleq1 2676 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 329 . . 3 (𝑦 = 𝐴 → ((𝜑𝑦𝑥) ↔ (𝜑𝐴𝑥)))
43ralbidv 2969 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 (𝜑𝑦𝑥) ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
5 vex 3176 . . 3 𝑦 ∈ V
65elintrab 4423 . 2 (𝑦 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝑦𝑥))
71, 4, 6vtoclbg 3240 1 (𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-int 4411 This theorem is referenced by:  tskmid  9541  eltskm  9544  ldsysgenld  29550  ldgenpisyslem1  29553  nobndlem6  31096  elpcliN  34197  elmapintrab  36901
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