Theorem sbcnestgf 3106

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 2962	. . . . 5 |- ( z = A -> ( [. z / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / y ]. ph ) )
2		csbeq1 3058	. . . . . 6 |- ( z = A -> [_ z / x ]_ B = [_ A / x ]_ B )
3		dfsbcq 2962	. . . . . 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 2738	. . . . 5 |- z e. _V
8	7	a1i 9	. . . 4 |- ( A. y F/ x ph -> z e. _V )
9		csbeq1a 3064	. . . . . 6 |- ( x = z -> B = [_ z / x ]_ B )
10		dfsbcq 2962	. . . . . 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 1542	. . . . 5 |- F/ x F/ x ph
14	13	nfal 1574	. . . 4 |- F/ x A. y F/ x ph
15		nfa1 1539	. . . . 5 |- F/ y A. y F/ x ph
16		nfcsb1v 3088	. . . . . 6 |- F/_ x [_ z / x ]_ B
17	16	a1i 9	. . . . 5 |- ( A. y F/ x ph -> F/_ x [_ z / x ]_ B )
18		sp 1509	. . . . 5 |- ( A. y F/ x ph -> F/ x ph )
19	15, 17, 18	nfsbcd 2980	. . . 4 |- ( A. y F/ x ph -> F/ x [. [_ z / x ]_ B / y ]. ph )
20	8, 12, 14, 19	sbciedf 2996	. . 3 |- ( A. y F/ x ph -> ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
21	6, 20	vtoclg 2795	. 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 1458   e. wcel 2146   F/_wnfc 2304   _Vcvv 2735   [.wsbc 2960   [_csb 3055

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-sbc 2961  df-csb 3056

This theorem is referenced by:  csbnestgf  3107  sbcnestg  3108