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Theorem frege54cor1c 37726
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 3202 . . . 4 𝐴 ∈ V
32snid 4184 . . 3 𝐴 ∈ {𝐴}
4 df-sn 4154 . . 3 {𝐴} = {𝑥𝑥 = 𝐴}
53, 4eleqtri 2696 . 2 𝐴 ∈ {𝑥𝑥 = 𝐴}
6 df-sbc 3422 . 2 ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 ∈ {𝑥𝑥 = 𝐴})
75, 6mpbir 221 1 [𝐴 / 𝑥]𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  {cab 2607  [wsbc 3421  {csn 4153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-sbc 3422  df-sn 4154
This theorem is referenced by:  frege55lem2c  37728  frege55c  37729  frege56c  37730
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