Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  abeq1i GIF version

Theorem abeq1i 2165
 Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
Hypothesis
Ref Expression
abeqri.1 {𝑥𝜑} = 𝐴
Assertion
Ref Expression
abeq1i (𝜑𝑥𝐴)

Proof of Theorem abeq1i
StepHypRef Expression
1 abid 2044 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
2 abeqri.1 . . 3 {𝑥𝜑} = 𝐴
32eleq2i 2120 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝑥𝐴)
41, 3bitr3i 179 1 (𝜑𝑥𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 102   = wceq 1259   ∈ wcel 1409  {cab 2042 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 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator