Theorem sbcnestgf 3145

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 3000	. . . . 5 |- ( z = A -> ( [. z / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / y ]. ph ) )
2		csbeq1 3096	. . . . . 6 |- ( z = A -> [_ z / x ]_ B = [_ A / x ]_ B )
3		dfsbcq 3000	. . . . . 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 2775	. . . . 5 |- z e. _V
8	7	a1i 9	. . . 4 |- ( A. y F/ x ph -> z e. _V )
9		csbeq1a 3102	. . . . . 6 |- ( x = z -> B = [_ z / x ]_ B )
10		dfsbcq 3000	. . . . . 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 1567	. . . . 5 |- F/ x F/ x ph
14	13	nfal 1599	. . . 4 |- F/ x A. y F/ x ph
15		nfa1 1564	. . . . 5 |- F/ y A. y F/ x ph
16		nfcsb1v 3126	. . . . . 6 |- F/_ x [_ z / x ]_ B
17	16	a1i 9	. . . . 5 |- ( A. y F/ x ph -> F/_ x [_ z / x ]_ B )
18		sp 1534	. . . . 5 |- ( A. y F/ x ph -> F/ x ph )
19	15, 17, 18	nfsbcd 3018	. . . 4 |- ( A. y F/ x ph -> F/ x [. [_ z / x ]_ B / y ]. ph )
20	8, 12, 14, 19	sbciedf 3034	. . 3 |- ( A. y F/ x ph -> ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
21	6, 20	vtoclg 2833	. 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 1371   = wceq 1373   F/wnf 1483   e. wcel 2176   F/_wnfc 2335   _Vcvv 2772   [.wsbc 2998   [_csb 3093

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999  df-csb 3094

This theorem is referenced by:  csbnestgf  3146  sbcnestg  3147