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

Theorem eqvisset 2582
 Description: A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2578 and issetri 2581. (Contributed by BJ, 27-Apr-2019.)
Assertion
Ref Expression
eqvisset (𝑥 = 𝐴𝐴 ∈ V)

Proof of Theorem eqvisset
StepHypRef Expression
1 vex 2577 . 2 𝑥 ∈ V
2 eleq1 2116 . 2 (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
31, 2mpbii 140 1 (𝑥 = 𝐴𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   ∈ wcel 1409  Vcvv 2574 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576 This theorem is referenced by:  xpsnen  6325
 Copyright terms: Public domain W3C validator