Theorem csbidmg 2958

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

Assertion

Ref	Expression
csbidmg	|- ( A e. V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   V( x)

Proof of Theorem csbidmg

Step	Hyp	Ref	Expression
1		elex 2610	. 2 |- ( A e. V -> A e. _V )
2		csbnest1g 2957	. . 3 |- ( A e. _V -> [_ A / x ]_ [_ A / x ]_ B = [_ [_ A / x ]_ A / x ]_ B )
3		csbconstg 2920	. . . 4 |- ( A e. _V -> [_ A / x ]_ A = A )
4	3	csbeq1d 2914	. . 3 |- ( A e. _V -> [_ [_ A / x ]_ A / x ]_ B = [_ A / x ]_ B )
5	2, 4	eqtrd 2113	. 2 |- ( A e. _V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )
6	1, 5	syl 14	1 |- ( A e. V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   _Vcvv 2601   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by: (None)