Theorem bj-csbsn 37149

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 37148	. . 3 ⊢ ⦋𝑦 / 𝑥⦌{𝑥} = {𝑦}
2	1	csbeq2i 3859	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑦⦌{𝑦}
3		csbcow 3866	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑥⦌{𝑥}
4		bj-csbsnlem 37148	. 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴}
5	2, 3, 4	3eqtr3i 2768	1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}

Syntax hints:   = wceq 1542  ⦋csb 3851  {csn 4582

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-sbc 3743  df-csb 3852  df-sn 4583

This theorem is referenced by:  bj-snsetex  37208