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Theorem frege54cor1c 38736
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 3365 . . . 4 𝐴 ∈ V
32snid 4348 . . 3 𝐴 ∈ {𝐴}
4 df-sn 4318 . . 3 {𝐴} = {𝑥𝑥 = 𝐴}
53, 4eleqtri 2848 . 2 𝐴 ∈ {𝑥𝑥 = 𝐴}
6 df-sbc 3589 . 2 ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 ∈ {𝑥𝑥 = 𝐴})
75, 6mpbir 221 1 [𝐴 / 𝑥]𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  {cab 2757  [wsbc 3588  {csn 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3589  df-sn 4318
This theorem is referenced by:  frege55lem2c  38738  frege55c  38739  frege56c  38740
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