Theorem frege54cor1c 41152

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 3420	. . . 4 ⊢ 𝐴 ∈ V
3	2	snid 4567	. . 3 ⊢ 𝐴 ∈ {𝐴}
4		df-sn 4532	. . 3 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
5	3, 4	eleqtri 2832	. 2 ⊢ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴}
6		df-sbc 3688	. 2 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴})
7	5, 6	mpbir 234	1 ⊢ [𝐴 / 𝑥]𝑥 = 𝐴

Syntax hints:   = wceq 1543   ∈ wcel 2110  {cab 2712  [wsbc 3687  {csn 4531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2706

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-v 3403  df-sbc 3688  df-sn 4532

This theorem is referenced by:  frege55lem2c  41154  frege55c  41155  frege56c  41156