Theorem frege54cor1c 42129

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 3462	. . . 4 ⊢ 𝐴 ∈ V
3	2	snid 4620	. . 3 ⊢ 𝐴 ∈ {𝐴}
4		df-sn 4585	. . 3 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
5	3, 4	eleqtri 2836	. 2 ⊢ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴}
6		df-sbc 3738	. 2 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴})
7	5, 6	mpbir 230	1 ⊢ [𝐴 / 𝑥]𝑥 = 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2106  {cab 2713  [wsbc 3737  {csn 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3445  df-sbc 3738  df-sn 4585

This theorem is referenced by:  frege55lem2c  42131  frege55c  42132  frege56c  42133