Theorem csbnestgf 3200

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 2833	. . 3 |- ( A e. V -> A e. _V )
2		df-csb 3148	. . . . . . 7 |- [_ B / y ]_ C = { z | [. B / y ]. z e. C }
3	2	abeq2i 2349	. . . . . 6 |- ( z e. [_ B / y ]_ C <-> [. B / y ]. z e. C )
4	3	sbcbii 3111	. . . . 5 |- ( [. A / x ]. z e. [_ B / y ]_ C <-> [. A / x ]. [. B / y ]. z e. C )
5		nfcr 2384	. . . . . . 7 |- ( F/_ x C -> F/ x z e. C )
6	5	alimi 1508	. . . . . 6 |- ( A. y F/_ x C -> A. y F/ x z e. C )
7		sbcnestgf 3199	. . . . . 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 2358	. . 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 3148	. 2 |- [_ A / x ]_ [_ B / y ]_ C = { z | [. A / x ]. z e. [_ B / y ]_ C }
13		df-csb 3148	. 2 |- [_ [_ A / x ]_ B / y ]_ C = { z | [. [_ A / x ]_ B / y ]. z e. C }
14	11, 12, 13	3eqtr4g 2296	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 1400   = wceq 1402   F/wnf 1513   e. wcel 2209   {cab 2224   F/_wnfc 2379   _Vcvv 2821   [.wsbc 3051   [_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:  csbnestg  3202  csbnest1g  3203