Theorem csbid 3850

Description: Analogue of sbid 2263 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 3838	. 2 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴}
2		sbcid 3745	. . 3 ⊢ ([𝑥 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
3	2	abbii 2803	. 2 ⊢ {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴}
4		abid2 2873	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
5	1, 3, 4	3eqtri 2763	1 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2114  {cab 2714  [wsbc 3728  ⦋csb 3837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-sbc 3729  df-csb 3838

This theorem is referenced by:  csbeq1a  3851  fvmpt2f  6948  fvmpt2i  6958  fvmpocurryd  8221  fsumsplitf  15704  gsummoncoe1  22273  gsumply1eq  22274  disji2f  32647  disjif2  32651  disjabrex  32652  disjabrexf  32653  gsummpt2co  33109  measiuns  34361  fphpd  43244  disjrnmpt2  45618  climinf2mpt  46142  climinfmpt  46143  dvmptmulf  46365  sge0f1o  46810