Theorem csbnest1g 3203

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 3180	. . . 4 |- F/_ x [_ y / x ]_ C
2	1	ax-gen 1502	. . 3 |- A. y F/_ x [_ y / x ]_ C
3		csbnestgf 3200	. . 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 429	. 2 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ [_ y / x ]_ C = [_ [_ A / x ]_ B / y ]_ [_ y / x ]_ C )
5		csbco 3157	. . 3 |- [_ B / y ]_ [_ y / x ]_ C = [_ B / x ]_ C
6	5	csbeq2i 3174	. 2 |- [_ A / x ]_ [_ B / y ]_ [_ y / x ]_ C = [_ A / x ]_ [_ B / x ]_ C
7		csbco 3157	. 2 |- [_ [_ A / x ]_ B / y ]_ [_ y / x ]_ C = [_ [_ A / x ]_ B / x ]_ C
8	4, 6, 7	3eqtr3g 2294	1 |- ( A e. V -> [_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C )

Syntax hints:   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   F/_wnfc 2379   [_csb 3147

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148

This theorem is referenced by:  csbidmg  3204