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Theorem frege54cor1c 37229
 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 3186 . . . 4 𝐴 ∈ V
32snid 4155 . . 3 𝐴 ∈ {𝐴}
4 df-sn 4126 . . 3 {𝐴} = {𝑥𝑥 = 𝐴}
53, 4eleqtri 2686 . 2 𝐴 ∈ {𝑥𝑥 = 𝐴}
6 df-sbc 3403 . 2 ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 ∈ {𝑥𝑥 = 𝐴})
75, 6mpbir 220 1 [𝐴 / 𝑥]𝑥 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  {cab 2596  [wsbc 3402  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-sn 4126 This theorem is referenced by:  frege55lem2c  37231  frege55c  37232  frege56c  37233
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