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

Theorem zfauscl 3919
Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3917, we invoke the Axiom of Extensionality (indirectly via vtocl 2663), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
zfauscl.1 𝐴 ∈ V
Assertion
Ref Expression
zfauscl 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem zfauscl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 zfauscl.1 . 2 𝐴 ∈ V
2 eleq2 2146 . . . . . 6 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
32anbi1d 453 . . . . 5 (𝑧 = 𝐴 → ((𝑥𝑧𝜑) ↔ (𝑥𝐴𝜑)))
43bibi2d 230 . . . 4 (𝑧 = 𝐴 → ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ (𝑥𝑦 ↔ (𝑥𝐴𝜑))))
54albidv 1747 . . 3 (𝑧 = 𝐴 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
65exbidv 1748 . 2 (𝑧 = 𝐴 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))))
7 ax-sep 3917 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
81, 6, 7vtocl 2663 1 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434  Vcvv 2611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065  ax-sep 3917
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2613
This theorem is referenced by:  inex1  3933  bj-d0clsepcl  11012
  Copyright terms: Public domain W3C validator