Theorem bj-csbsn 35089

Description: Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-csbsn	⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}

Proof of Theorem bj-csbsn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-csbsnlem 35088	. . 3 ⊢ ⦋𝑦 / 𝑥⦌{𝑥} = {𝑦}
2	1	csbeq2i 3840	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑦⦌{𝑦}
3		csbcow 3847	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑥⦌{𝑥}
4		bj-csbsnlem 35088	. 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴}
5	2, 3, 4	3eqtr3i 2774	1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}

Syntax hints:   = wceq 1539  ⦋csb 3832  {csn 4561

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-sbc 3717  df-csb 3833  df-sn 4562

This theorem is referenced by:  bj-snsetex  35153