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Theorem elintrabg 4965
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 2820 . 2 (𝑦 = 𝐴 → (𝑦 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥𝐵𝜑}))
2 eleq1 2820 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 340 . . 3 (𝑦 = 𝐴 → ((𝜑𝑦𝑥) ↔ (𝜑𝐴𝑥)))
43ralbidv 3176 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 (𝜑𝑦𝑥) ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
5 vex 3477 . . 3 𝑦 ∈ V
65elintrab 4964 . 2 (𝑦 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝑦𝑥))
71, 4, 6vtoclbg 3544 1 (𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  wral 3060  {crab 3431   cint 4950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-int 4951
This theorem is referenced by:  tskmid  10838  eltskm  10841  ldsysgenld  33457  ldgenpisyslem1  33460  elpcliN  39068  elmapintrab  42630
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