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Theorem frege54cor1c 39165
 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 3415 . . . 4 𝐴 ∈ V
32snid 4430 . . 3 𝐴 ∈ {𝐴}
4 df-sn 4399 . . 3 {𝐴} = {𝑥𝑥 = 𝐴}
53, 4eleqtri 2857 . 2 𝐴 ∈ {𝑥𝑥 = 𝐴}
6 df-sbc 3653 . 2 ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 ∈ {𝑥𝑥 = 𝐴})
75, 6mpbir 223 1 [𝐴 / 𝑥]𝑥 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601   ∈ wcel 2107  {cab 2763  [wsbc 3652  {csn 4398 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-sbc 3653  df-sn 4399 This theorem is referenced by:  frege55lem2c  39167  frege55c  39168  frege56c  39169
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