Theorem csbid 3144

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 3138	. 2 |- [_x / x]_A = {y | [.x / x ].y e. A}
2		sbsbc 3051	. . . 4 |- ([x / x]y e. A <-> [.x / x ].y e. A)
3		sbid 1922	. . . 4 |- ([x / x]y e. A <-> y e. A)
4	2, 3	bitr3i 242	. . 3 |- ( [.x / x ].y e. A <-> y e. A)
5	4	abbii 2466	. 2 |- {y | [.x / x ].y e. A} = {y | y e. A}
6		abid2 2471	. 2 |- {y | y e. A} = A
7	1, 5, 6	3eqtri 2377	1 |- [_x / x]_A = A

Syntax hints:   = wceq 1642  [wsb 1648   e. wcel 1710  {cab 2339   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 3048  df-csb 3138

This theorem is referenced by:  csbeq1a  3145  fvmpt2i  5704