Theorem sbcnestgf 3100

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 2957	. . . . 5 |- ( z = A -> ( [. z / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / y ]. ph ) )
2		csbeq1 3052	. . . . . 6 |- ( z = A -> [_ z / x ]_ B = [_ A / x ]_ B )
3		dfsbcq 2957	. . . . . 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 234	. . . 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 229	. . 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 2733	. . . . 5 |- z e. _V
8	7	a1i 9	. . . 4 |- ( A. y F/ x ph -> z e. _V )
9		csbeq1a 3058	. . . . . 6 |- ( x = z -> B = [_ z / x ]_ B )
10		dfsbcq 2957	. . . . . 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 275	. . . 4 |- ( ( A. y F/ x ph /\ x = z ) -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
13		nfnf1 1537	. . . . 5 |- F/ x F/ x ph
14	13	nfal 1569	. . . 4 |- F/ x A. y F/ x ph
15		nfa1 1534	. . . . 5 |- F/ y A. y F/ x ph
16		nfcsb1v 3082	. . . . . 6 |- F/_ x [_ z / x ]_ B
17	16	a1i 9	. . . . 5 |- ( A. y F/ x ph -> F/_ x [_ z / x ]_ B )
18		sp 1504	. . . . 5 |- ( A. y F/ x ph -> F/ x ph )
19	15, 17, 18	nfsbcd 2974	. . . 4 |- ( A. y F/ x ph -> F/ x [. [_ z / x ]_ B / y ]. ph )
20	8, 12, 14, 19	sbciedf 2990	. . 3 |- ( A. y F/ x ph -> ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
21	6, 20	vtoclg 2790	. 2 |- ( A e. V -> ( A. y F/ x ph -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) )
22	21	imp 123	1 |- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   F/wnf 1453   e. wcel 2141   F/_wnfc 2299   _Vcvv 2730   [.wsbc 2955   [_csb 3049

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  csbnestgf  3101  sbcnestg  3102