Theorem csbid 3921

Description: Analogue of sbid 2253 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 3909	. 2 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴}
2		sbcid 3808	. . 3 ⊢ ([𝑥 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
3	2	abbii 2807	. 2 ⊢ {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴}
4		abid2 2877	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
5	1, 3, 4	3eqtri 2767	1 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2106  {cab 2712  [wsbc 3791  ⦋csb 3908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792  df-csb 3909

This theorem is referenced by:  csbeq1a  3922  fvmpt2f  7017  fvmpt2i  7026  fvmpocurryd  8295  fsumsplitf  15775  gsummoncoe1  22328  gsumply1eq  22329  disji2f  32597  disjif2  32601  disjabrex  32602  disjabrexf  32603  gsummpt2co  33034  measiuns  34198  fphpd  42804  disjrnmpt2  45131  climinf2mpt  45670  climinfmpt  45671  dvmptmulf  45893  sge0f1o  46338