Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elintrabg Structured version   Visualization version   GIF version

Theorem elintrabg 4854
 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 2839 . 2 (𝑦 = 𝐴 → (𝑦 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥𝐵𝜑}))
2 eleq1 2839 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 344 . . 3 (𝑦 = 𝐴 → ((𝜑𝑦𝑥) ↔ (𝜑𝐴𝑥)))
43ralbidv 3126 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 (𝜑𝑦𝑥) ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
5 vex 3413 . . 3 𝑦 ∈ V
65elintrab 4853 . 2 (𝑦 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝑦𝑥))
71, 4, 6vtoclbg 3489 1 (𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  ∩ cint 4841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rab 3079  df-v 3411  df-int 4842 This theorem is referenced by:  tskmid  10313  eltskm  10316  ldsysgenld  31660  ldgenpisyslem1  31663  elpcliN  37504  elmapintrab  40694
 Copyright terms: Public domain W3C validator