Theorem csbnest1g 3112

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 3090	. . . 4 |- F/_ x [_ y / x ]_ C
2	1	ax-gen 1449	. . 3 |- A. y F/_ x [_ y / x ]_ C
3		csbnestgf 3109	. . 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 425	. 2 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ [_ y / x ]_ C = [_ [_ A / x ]_ B / y ]_ [_ y / x ]_ C )
5		csbco 3067	. . 3 |- [_ B / y ]_ [_ y / x ]_ C = [_ B / x ]_ C
6	5	csbeq2i 3084	. 2 |- [_ A / x ]_ [_ B / y ]_ [_ y / x ]_ C = [_ A / x ]_ [_ B / x ]_ C
7		csbco 3067	. 2 |- [_ [_ A / x ]_ B / y ]_ [_ y / x ]_ C = [_ [_ A / x ]_ B / x ]_ C
8	4, 6, 7	3eqtr3g 2233	1 |- ( A e. V -> [_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C )

Syntax hints:   -> wi 4   A.wal 1351   = wceq 1353   e. wcel 2148   F/_wnfc 2306   [_csb 3057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-csb 3058

This theorem is referenced by:  csbidmg  3113