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Theorem frege54cor1c 43406
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 3484 . . . 4 𝐴 ∈ V
32snid 4661 . . 3 𝐴 ∈ {𝐴}
4 df-sn 4626 . . 3 {𝐴} = {𝑥𝑥 = 𝐴}
53, 4eleqtri 2823 . 2 𝐴 ∈ {𝑥𝑥 = 𝐴}
6 df-sbc 3771 . 2 ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 ∈ {𝑥𝑥 = 𝐴})
75, 6mpbir 230 1 [𝐴 / 𝑥]𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  {cab 2702  [wsbc 3770  {csn 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-sbc 3771  df-sn 4626
This theorem is referenced by:  frege55lem2c  43408  frege55c  43409  frege56c  43410
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