Theorem sbcnestgf 3176

Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)

Assertion

Ref	Expression
sbcnestgf	|- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )

Proof of Theorem sbcnestgf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3030	. . . . 5 |- ( z = A -> ( [. z / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / y ]. ph ) )
2		csbeq1 3127	. . . . . 6 |- ( z = A -> [_ z / x ]_ B = [_ A / x ]_ B )
3		dfsbcq 3030	. . . . . 6 |- ( [_ z / x ]_ B = [_ A / x ]_ B -> ( [. [_ z / x ]_ B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
4	2, 3	syl 14	. . . . 5 |- ( z = A -> ( [. [_ z / x ]_ B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
5	1, 4	bibi12d 235	. . . 4 |- ( z = A -> ( ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) <-> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) )
6	5	imbi2d 230	. . 3 |- ( z = A -> ( ( A. y F/ x ph -> ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) ) <-> ( A. y F/ x ph -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) ) )
7		vex 2802	. . . . 5 |- z e. _V
8	7	a1i 9	. . . 4 |- ( A. y F/ x ph -> z e. _V )
9		csbeq1a 3133	. . . . . 6 |- ( x = z -> B = [_ z / x ]_ B )
10		dfsbcq 3030	. . . . . 6 |- ( B = [_ z / x ]_ B -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
11	9, 10	syl 14	. . . . 5 |- ( x = z -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
12	11	adantl 277	. . . 4 |- ( ( A. y F/ x ph /\ x = z ) -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
13		nfnf1 1590	. . . . 5 |- F/ x F/ x ph
14	13	nfal 1622	. . . 4 |- F/ x A. y F/ x ph
15		nfa1 1587	. . . . 5 |- F/ y A. y F/ x ph
16		nfcsb1v 3157	. . . . . 6 |- F/_ x [_ z / x ]_ B
17	16	a1i 9	. . . . 5 |- ( A. y F/ x ph -> F/_ x [_ z / x ]_ B )
18		sp 1557	. . . . 5 |- ( A. y F/ x ph -> F/ x ph )
19	15, 17, 18	nfsbcd 3048	. . . 4 |- ( A. y F/ x ph -> F/ x [. [_ z / x ]_ B / y ]. ph )
20	8, 12, 14, 19	sbciedf 3064	. . 3 |- ( A. y F/ x ph -> ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
21	6, 20	vtoclg 2861	. 2 |- ( A e. V -> ( A. y F/ x ph -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) )
22	21	imp 124	1 |- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   F/wnf 1506   e. wcel 2200   F/_wnfc 2359   _Vcvv 2799   [.wsbc 3028   [_csb 3124

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by:  csbnestgf  3177  sbcnestg  3178