Theorem csbnestgf 3996

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 3212	. . 3 |- ( A e. V -> A e. _V )
2		df-csb 3534	. . . . . . 7 |- [_ B / y ]_ C = { z | [. B / y ]. z e. C }
3	2	abeq2i 2735	. . . . . 6 |- ( z e. [_ B / y ]_ C <-> [. B / y ]. z e. C )
4	3	sbcbii 3491	. . . . 5 |- ( [. A / x ]. z e. [_ B / y ]_ C <-> [. A / x ]. [. B / y ]. z e. C )
5		nfcr 2756	. . . . . . 7 |- ( F/_ x C -> F/ x z e. C )
6	5	alimi 1739	. . . . . 6 |- ( A. y F/_ x C -> A. y F/ x z e. C )
7		sbcnestgf 3995	. . . . . 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 491	. . . . 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 272	. . . 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 2741	. . 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 488	. 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 3534	. 2 |- [_ A / x ]_ [_ B / y ]_ C = { z | [. A / x ]. z e. [_ B / y ]_ C }
13		df-csb 3534	. 2 |- [_ [_ A / x ]_ B / y ]_ C = { z | [. [_ A / x ]_ B / y ]. z e. C }
14	11, 12, 13	3eqtr4g 2681	1 |- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   {cab 2608   F/_wnfc 2751   _Vcvv 3200   [.wsbc 3435   [_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:  csbnestg  3998  csbnest1g  4001