Theorem bj-csbsn 37349

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 37348	. . 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 37348	. 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴}
5	2, 3, 4	3eqtr3i 2792	1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴}

Syntax hints:   = wceq 1559  ⦋csb 3850  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-sbc 3743  df-csb 3851  df-sn 4580

This theorem is referenced by:  bj-snsetex  37408