Theorem csbnestgf 3079

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 2720	. . 3 |- ( A e. V -> A e. _V )
2		df-csb 3028	. . . . . . 7 |- [_ B / y ]_ C = { z | [. B / y ]. z e. C }
3	2	abeq2i 2265	. . . . . 6 |- ( z e. [_ B / y ]_ C <-> [. B / y ]. z e. C )
4	3	sbcbii 2992	. . . . 5 |- ( [. A / x ]. z e. [_ B / y ]_ C <-> [. A / x ]. [. B / y ]. z e. C )
5		nfcr 2288	. . . . . . 7 |- ( F/_ x C -> F/ x z e. C )
6	5	alimi 1432	. . . . . 6 |- ( A. y F/_ x C -> A. y F/ x z e. C )
7		sbcnestgf 3078	. . . . . 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 284	. . . . 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 191	. . . 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 2272	. . 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 281	. 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 3028	. 2 |- [_ A / x ]_ [_ B / y ]_ C = { z | [. A / x ]. z e. [_ B / y ]_ C }
13		df-csb 3028	. 2 |- [_ [_ A / x ]_ B / y ]_ C = { z | [. [_ A / x ]_ B / y ]. z e. C }
14	11, 12, 13	3eqtr4g 2212	1 |- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   = wceq 1332   F/wnf 1437   e. wcel 2125   {cab 2140   F/_wnfc 2283   _Vcvv 2709   [.wsbc 2933   [_csb 3027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-sbc 2934  df-csb 3028

This theorem is referenced by:  csbnestg  3081  csbnest1g  3082