Theorem csbid 3904

Description: Analogue of sbid 2243 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 3892	. 2 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴}
2		sbcid 3792	. . 3 ⊢ ([𝑥 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
3	2	abbii 2796	. 2 ⊢ {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴}
4		abid2 2864	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
5	1, 3, 4	3eqtri 2758	1 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴

Syntax hints:   = wceq 1534   ∈ wcel 2099  {cab 2703  [wsbc 3775  ⦋csb 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-sbc 3776  df-csb 3892

This theorem is referenced by:  csbeq1a  3905  fvmpt2f  7002  fvmpt2i  7011  fvmpocurryd  8278  fsumsplitf  15741  gsummoncoe1  22296  gsumply1eq  22297  disji2f  32497  disjif2  32501  disjabrex  32502  disjabrexf  32503  gsummpt2co  32920  measiuns  34063  fphpd  42510  disjrnmpt2  44831  climinf2mpt  45371  climinfmpt  45372  dvmptmulf  45594  sge0f1o  46039