Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  abeq1i Structured version   Visualization version   GIF version

Theorem abeq1i 2950
 Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Nov-2019.)
Hypothesis
Ref Expression
abeq1i.1 {𝑥𝜑} = 𝐴
Assertion
Ref Expression
abeq1i (𝜑𝑥𝐴)

Proof of Theorem abeq1i
StepHypRef Expression
1 abeq1i.1 . . . 4 {𝑥𝜑} = 𝐴
21eqcomi 2831 . . 3 𝐴 = {𝑥𝜑}
32abeq2i 2949 . 2 (𝑥𝐴𝜑)
43bicomi 227 1 (𝜑𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2114  {cab 2800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator