Theorem csbid 3143

Description: Analog of sbid 1922 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 3137	. 2 ⊢ [x / x]A = {y ∣ [̣x / x]̣y ∈ A}
2		sbsbc 3050	. . . 4 ⊢ ([x / x]y ∈ A ↔ [̣x / x]̣y ∈ A)
3		sbid 1922	. . . 4 ⊢ ([x / x]y ∈ A ↔ y ∈ A)
4	2, 3	bitr3i 242	. . 3 ⊢ ([̣x / x]̣y ∈ A ↔ y ∈ A)
5	4	abbii 2465	. 2 ⊢ {y ∣ [̣x / x]̣y ∈ A} = {y ∣ y ∈ A}
6		abid2 2470	. 2 ⊢ {y ∣ y ∈ A} = A
7	1, 5, 6	3eqtri 2377	1 ⊢ [x / x]A = A

Syntax hints:   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3047  df-csb 3137

This theorem is referenced by:  csbeq1a  3144  fvmpt2i  5703