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Theorem frege54cor1c 42261
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 3467 . . . 4 𝐴 ∈ V
32snid 4627 . . 3 𝐴 ∈ {𝐴}
4 df-sn 4592 . . 3 {𝐴} = {𝑥𝑥 = 𝐴}
53, 4eleqtri 2836 . 2 𝐴 ∈ {𝑥𝑥 = 𝐴}
6 df-sbc 3745 . 2 ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 ∈ {𝑥𝑥 = 𝐴})
75, 6mpbir 230 1 [𝐴 / 𝑥]𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  {cab 2714  [wsbc 3744  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-sbc 3745  df-sn 4592
This theorem is referenced by:  frege55lem2c  42263  frege55c  42264  frege56c  42265
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