NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  abeq1i GIF version

Theorem abeq1i 2461
Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
Hypothesis
Ref Expression
abeqri.1 {x φ} = A
Assertion
Ref Expression
abeq1i (φx A)

Proof of Theorem abeq1i
StepHypRef Expression
1 abid 2341 . 2 (x {x φ} ↔ φ)
2 abeqri.1 . . 3 {x φ} = A
32eleq2i 2417 . 2 (x {x φ} ↔ x A)
41, 3bitr3i 242 1 (φx A)
Colors of variables: wff setvar class
Syntax hints:  wb 176   = wceq 1642   wcel 1710  {cab 2339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator