Theorem bj-csbsn 37655

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 37654	. . 3 ⊢ ⦋𝑦 / 𝑥⦌{𝑥} = {𝑦}
2	1	csbeq2i 3858	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑦⦌{𝑦}
3		csbcow 3865	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑥⦌{𝑥}
4		bj-csbsnlem 37654	. 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴}
5	2, 3, 4	3eqtr3i 2793	1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}

This proof depends on syntax axioms:   = wceq 1570  ⦋csb 3850  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sbc 3743  df-csb 3851  df-sn 4588

This theorem is used by:  bj-snsetex  37715