Theorem frege54cor1c 39165

Description: Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)

Hypothesis

Ref	Expression
frege54c.1	⊢ 𝐴 ∈ 𝐶

Assertion

Ref	Expression
frege54cor1c	⊢ [𝐴 / 𝑥]𝑥 = 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem frege54cor1c

Step	Hyp	Ref	Expression
1		frege54c.1	. . . . 5 ⊢ 𝐴 ∈ 𝐶
2	1	elexi 3415	. . . 4 ⊢ 𝐴 ∈ V
3	2	snid 4430	. . 3 ⊢ 𝐴 ∈ {𝐴}
4		df-sn 4399	. . 3 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
5	3, 4	eleqtri 2857	. 2 ⊢ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴}
6		df-sbc 3653	. 2 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴})
7	5, 6	mpbir 223	1 ⊢ [𝐴 / 𝑥]𝑥 = 𝐴

Syntax hints:   = wceq 1601   ∈ wcel 2107  {cab 2763  [wsbc 3652  {csn 4398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-sbc 3653  df-sn 4399

This theorem is referenced by:  frege55lem2c  39167  frege55c  39168  frege56c  39169