Theorem csbnest1g 4001

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 3549	. . . 4 |- F/_ x [_ y / x ]_ C
2	1	ax-gen 1722	. . 3 |- A. y F/_ x [_ y / x ]_ C
3		csbnestgf 3996	. . 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 707	. 2 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ [_ y / x ]_ C = [_ [_ A / x ]_ B / y ]_ [_ y / x ]_ C )
5		csbco 3543	. . 3 |- [_ B / y ]_ [_ y / x ]_ C = [_ B / x ]_ C
6	5	csbeq2i 3993	. 2 |- [_ A / x ]_ [_ B / y ]_ [_ y / x ]_ C = [_ A / x ]_ [_ B / x ]_ C
7		csbco 3543	. 2 |- [_ [_ A / x ]_ B / y ]_ [_ y / x ]_ C = [_ [_ A / x ]_ B / x ]_ C
8	4, 6, 7	3eqtr3g 2679	1 |- ( A e. V -> [_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   F/_wnfc 2751   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbidm  4002