Theorem csbid 3135

Description: Analog of sbid 1822 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbid	⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴

Proof of Theorem csbid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3128	. 2 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴}
2		sbcid 3047	. . 3 ⊢ ([𝑥 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
3	2	abbii 2347	. 2 ⊢ {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴}
4		abid2 2352	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
5	1, 3, 4	3eqtri 2256	1 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴

Syntax hints:   = wceq 1397   ∈ wcel 2202  {cab 2217  [wsbc 3031  ⦋csb 3127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-sbc 3032  df-csb 3128

This theorem is referenced by:  csbeq1a  3136  fvmpt2  5730  fsumsplitf  11971  ctiunctlemfo  13062