Theorem csbnestgf 3145

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 2782	. . 3 |- ( A e. V -> A e. _V )
2		df-csb 3093	. . . . . . 7 |- [_ B / y ]_ C = { z | [. B / y ]. z e. C }
3	2	abeq2i 2315	. . . . . 6 |- ( z e. [_ B / y ]_ C <-> [. B / y ]. z e. C )
4	3	sbcbii 3057	. . . . 5 |- ( [. A / x ]. z e. [_ B / y ]_ C <-> [. A / x ]. [. B / y ]. z e. C )
5		nfcr 2339	. . . . . . 7 |- ( F/_ x C -> F/ x z e. C )
6	5	alimi 1477	. . . . . 6 |- ( A. y F/_ x C -> A. y F/ x z e. C )
7		sbcnestgf 3144	. . . . . 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 286	. . . . 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	bitrid 192	. . . 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 2322	. . 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 283	. 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 3093	. 2 |- [_ A / x ]_ [_ B / y ]_ C = { z | [. A / x ]. z e. [_ B / y ]_ C }
13		df-csb 3093	. 2 |- [_ [_ A / x ]_ B / y ]_ C = { z | [. [_ A / x ]_ B / y ]. z e. C }
14	11, 12, 13	3eqtr4g 2262	1 |- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1370   = wceq 1372   F/wnf 1482   e. wcel 2175   {cab 2190   F/_wnfc 2334   _Vcvv 2771   [.wsbc 2997   [_csb 3092

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998  df-csb 3093

This theorem is referenced by:  csbnestg  3147  csbnest1g  3148