Theorem sbcnestgf 3110

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 2966	. . . . 5 |- ( z = A -> ( [. z / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / y ]. ph ) )
2		csbeq1 3062	. . . . . 6 |- ( z = A -> [_ z / x ]_ B = [_ A / x ]_ B )
3		dfsbcq 2966	. . . . . 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 2742	. . . . 5 |- z e. _V
8	7	a1i 9	. . . 4 |- ( A. y F/ x ph -> z e. _V )
9		csbeq1a 3068	. . . . . 6 |- ( x = z -> B = [_ z / x ]_ B )
10		dfsbcq 2966	. . . . . 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 1544	. . . . 5 |- F/ x F/ x ph
14	13	nfal 1576	. . . 4 |- F/ x A. y F/ x ph
15		nfa1 1541	. . . . 5 |- F/ y A. y F/ x ph
16		nfcsb1v 3092	. . . . . 6 |- F/_ x [_ z / x ]_ B
17	16	a1i 9	. . . . 5 |- ( A. y F/ x ph -> F/_ x [_ z / x ]_ B )
18		sp 1511	. . . . 5 |- ( A. y F/ x ph -> F/ x ph )
19	15, 17, 18	nfsbcd 2984	. . . 4 |- ( A. y F/ x ph -> F/ x [. [_ z / x ]_ B / y ]. ph )
20	8, 12, 14, 19	sbciedf 3000	. . 3 |- ( A. y F/ x ph -> ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
21	6, 20	vtoclg 2799	. 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 1351   = wceq 1353   F/wnf 1460   e. wcel 2148   F/_wnfc 2306   _Vcvv 2739   [.wsbc 2964   [_csb 3059

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965  df-csb 3060

This theorem is referenced by:  csbnestgf  3111  sbcnestg  3112