Theorem csbnestgf 2898

Description: Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Assertion

Ref	Expression
csbnestgf	|- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )

Proof of Theorem csbnestgf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2566	. . 3 |- ( A e. V -> A e. _V )
2		df-csb 2853	. . . . . . 7 |- [_ B / y ]_ C = { z | [. B / y ]. z e. C }
3	2	abeq2i 2148	. . . . . 6 |- ( z e. [_ B / y ]_ C <-> [. B / y ]. z e. C )
4	3	sbcbii 2818	. . . . 5 |- ( [. A / x ]. z e. [_ B / y ]_ C <-> [. A / x ]. [. B / y ]. z e. C )
5		nfcr 2170	. . . . . . 7 |- ( F/_ x C -> F/ x z e. C )
6	5	alimi 1344	. . . . . 6 |- ( A. y F/_ x C -> A. y F/ x z e. C )
7		sbcnestgf 2897	. . . . . 6 |- ( ( A e. _V /\ A. y F/ x z e. C ) -> ( [. A / x ]. [. B / y ]. z e. C <-> [. [_ A / x ]_ B / y ]. z e. C ) )
8	6, 7	sylan2 270	. . . . 5 |- ( ( A e. _V /\ A. y F/_ x C ) -> ( [. A / x ]. [. B / y ]. z e. C <-> [. [_ A / x ]_ B / y ]. z e. C ) )
9	4, 8	syl5bb 181	. . . 4 |- ( ( A e. _V /\ A. y F/_ x C ) -> ( [. A / x ]. z e. [_ B / y ]_ C <-> [. [_ A / x ]_ B / y ]. z e. C ) )
10	9	abbidv 2155	. . 3 |- ( ( A e. _V /\ A. y F/_ x C ) -> { z | [. A / x ]. z e. [_ B / y ]_ C } = { z | [. [_ A / x ]_ B / y ]. z e. C } )
11	1, 10	sylan 267	. 2 |- ( ( A e. V /\ A. y F/_ x C ) -> { z | [. A / x ]. z e. [_ B / y ]_ C } = { z | [. [_ A / x ]_ B / y ]. z e. C } )
12		df-csb 2853	. 2 |- [_ A / x ]_ [_ B / y ]_ C = { z | [. A / x ]. z e. [_ B / y ]_ C }
13		df-csb 2853	. 2 |- [_ [_ A / x ]_ B / y ]_ C = { z | [. [_ A / x ]_ B / y ]. z e. C }
14	11, 12, 13	3eqtr4g 2097	1 |- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   F/wnf 1349   e. wcel 1393   {cab 2026   F/_wnfc 2165   _Vcvv 2557   [.wsbc 2764   [_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:  csbnestg  2900  csbnest1g  2901