Theorem csbid 3869

Description: Analogue of sbid 2294 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 3857	. 2 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴}
2		sbcid 3764	. . 3 ⊢ ([𝑥 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
3	2	abbii 2833	. 2 ⊢ {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴}
4		abid2 2903	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
5	1, 3, 4	3eqtri 2793	1 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {cab 2744  [wsbc 3747  ⦋csb 3856

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-sbc 3748  df-csb 3857

This theorem is used by:  csbeq1a  3870  fvmpt2f  6997  fvmpt2i  7007  fvmpocurryd  8276  fsumsplitf  15819  gsummoncoe1  22505  gsumply1eq  22506  disji2f  32959  disjif2  32963  disjabrex  32964  disjabrexf  32965  gsummpt2co  33399  measiuns  34639  fphpd  43584  disjrnmpt2  45947  climinf2mpt  46469  climinfmpt  46470  dvmptmulf  46692  sge0f1o  47137