Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege54cor1c Structured version   Visualization version   GIF version

Theorem frege54cor1c 40603
 Description: Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)
Hypothesis
Ref Expression
frege54c.1 𝐴𝐶
Assertion
Ref Expression
frege54cor1c [𝐴 / 𝑥]𝑥 = 𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem frege54cor1c
StepHypRef Expression
1 frege54c.1 . . . . 5 𝐴𝐶
21elexi 3463 . . . 4 𝐴 ∈ V
32snid 4564 . . 3 𝐴 ∈ {𝐴}
4 df-sn 4529 . . 3 {𝐴} = {𝑥𝑥 = 𝐴}
53, 4eleqtri 2891 . 2 𝐴 ∈ {𝑥𝑥 = 𝐴}
6 df-sbc 3724 . 2 ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 ∈ {𝑥𝑥 = 𝐴})
75, 6mpbir 234 1 [𝐴 / 𝑥]𝑥 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2112  {cab 2779  [wsbc 3723  {csn 4528 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 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-sbc 3724  df-sn 4529 This theorem is referenced by:  frege55lem2c  40605  frege55c  40606  frege56c  40607
 Copyright terms: Public domain W3C validator