Theorem csbidmg 2902

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 2566	. 2 |- ( A e. V -> A e. _V )
2		csbnest1g 2901	. . 3 |- ( A e. _V -> [_ A / x ]_ [_ A / x ]_ B = [_ [_ A / x ]_ A / x ]_ B )
3		csbconstg 2864	. . . 4 |- ( A e. _V -> [_ A / x ]_ A = A )
4	3	csbeq1d 2858	. . 3 |- ( A e. _V -> [_ [_ A / x ]_ A / x ]_ B = [_ A / x ]_ B )
5	2, 4	eqtrd 2072	. 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 1243   e. wcel 1393   _Vcvv 2557   [_csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by: (None)