Theorem csbid 3011

Description: Analog of sbid 1747 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 3004	. 2 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴}
2		sbcid 2924	. . 3 ⊢ ([𝑥 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
3	2	abbii 2255	. 2 ⊢ {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴}
4		abid2 2260	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
5	1, 3, 4	3eqtri 2164	1 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴

Syntax hints:   = wceq 1331   ∈ wcel 1480  {cab 2125  [wsbc 2909  ⦋csb 3003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-sbc 2910  df-csb 3004

This theorem is referenced by:  csbeq1a  3012  fvmpt2  5504  fsumsplitf  11177  ctiunctlemfo  11952