Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  elintrabg Structured version   GIF version

Theorem elintrabg 3619
 Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
Assertion
Ref Expression
elintrabg (A 𝑉 → (A {x Bφ} ↔ x B (φA x)))
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 2097 . 2 (y = A → (y {x Bφ} ↔ A {x Bφ}))
2 eleq1 2097 . . . 4 (y = A → (y xA x))
32imbi2d 219 . . 3 (y = A → ((φy x) ↔ (φA x)))
43ralbidv 2320 . 2 (y = A → (x B (φy x) ↔ x B (φA x)))
5 vex 2554 . . 3 y V
65elintrab 3618 . 2 (y {x Bφ} ↔ x B (φy x))
71, 4, 6vtoclbg 2608 1 (A 𝑉 → (A {x Bφ} ↔ x B (φA x)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {crab 2304  ∩ cint 3606 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-int 3607 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator