Theorem bj-csbsn 32899

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 32898	. . 3 |- [_ y / x ]_ { x } = { y }
2	1	csbeq2i 3993	. 2 |- [_ A / y ]_ [_ y / x ]_ { x } = [_ A / y ]_ { y }
3		csbco 3543	. 2 |- [_ A / y ]_ [_ y / x ]_ { x } = [_ A / x ]_ { x }
4		bj-csbsnlem 32898	. 2 |- [_ A / y ]_ { y } = { A }
5	2, 3, 4	3eqtr3i 2652	1 |- [_ A / x ]_ { x } = { A }

Syntax hints:   = wceq 1483   [_csb 3533   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-sn 4178

This theorem is referenced by:  bj-snsetex  32951