Theorem csbidm 4002

Description: Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
csbidm	|- [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem csbidm

Step	Hyp	Ref	Expression
1		csbnest1g 4001	. . 3 |- ( A e. _V -> [_ A / x ]_ [_ A / x ]_ B = [_ [_ A / x ]_ A / x ]_ B )
2		csbconstg 3546	. . . 4 |- ( A e. _V -> [_ A / x ]_ A = A )
3	2	csbeq1d 3540	. . 3 |- ( A e. _V -> [_ [_ A / x ]_ A / x ]_ B = [_ A / x ]_ B )
4	1, 3	eqtrd 2656	. 2 |- ( A e. _V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )
5		csbprc 3980	. . 3 |- ( -. A e. _V -> [_ A / x ]_ [_ A / x ]_ B = (/) )
6		csbprc 3980	. . 3 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
7	5, 6	eqtr4d 2659	. 2 |- ( -. A e. _V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )
8	4, 7	pm2.61i 176	1 |- [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)