Theorem csbnest1g 3100

Description: Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Assertion

Ref	Expression
csbnest1g	|- ( A e. V -> [_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C )

Proof of Theorem csbnest1g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcsb1v 3078	. . . 4 |- F/_ x [_ y / x ]_ C
2	1	ax-gen 1437	. . 3 |- A. y F/_ x [_ y / x ]_ C
3		csbnestgf 3097	. . 3 |- ( ( A e. V /\ A. y F/_ x [_ y / x ]_ C ) -> [_ A / x ]_ [_ B / y ]_ [_ y / x ]_ C = [_ [_ A / x ]_ B / y ]_ [_ y / x ]_ C )
4	2, 3	mpan2 422	. 2 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ [_ y / x ]_ C = [_ [_ A / x ]_ B / y ]_ [_ y / x ]_ C )
5		csbco 3055	. . 3 |- [_ B / y ]_ [_ y / x ]_ C = [_ B / x ]_ C
6	5	csbeq2i 3072	. 2 |- [_ A / x ]_ [_ B / y ]_ [_ y / x ]_ C = [_ A / x ]_ [_ B / x ]_ C
7		csbco 3055	. 2 |- [_ [_ A / x ]_ B / y ]_ [_ y / x ]_ C = [_ [_ A / x ]_ B / x ]_ C
8	4, 6, 7	3eqtr3g 2222	1 |- ( A e. V -> [_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C )

Syntax hints:   -> wi 4   A.wal 1341   = wceq 1343   e. wcel 2136   F/_wnfc 2295   [_csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  csbidmg  3101