Theorem bj-csbsn 36887

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 36886	. . 3 ⊢ ⦋𝑦 / 𝑥⦌{𝑥} = {𝑦}
2	1	csbeq2i 3872	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑦⦌{𝑦}
3		csbcow 3879	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑥⦌{𝑥}
4		bj-csbsnlem 36886	. 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴}
5	2, 3, 4	3eqtr3i 2761	1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}

Syntax hints:   = wceq 1540  ⦋csb 3864  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-sbc 3756  df-csb 3865  df-sn 4592

This theorem is referenced by:  bj-snsetex  36946