Theorem sbcnestgf 3995

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 3437	. . . . 5 |- ( z = A -> ( [. z / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / y ]. ph ) )
2		csbeq1 3536	. . . . . 6 |- ( z = A -> [_ z / x ]_ B = [_ A / x ]_ B )
3	2	sbceq1d 3440	. . . . 5 |- ( z = A -> ( [. [_ z / x ]_ B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
4	1, 3	bibi12d 335	. . . 4 |- ( z = A -> ( ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) <-> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) )
5	4	imbi2d 330	. . 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 ) ) ) )
6		vex 3203	. . . . 5 |- z e. _V
7	6	a1i 11	. . . 4 |- ( A. y F/ x ph -> z e. _V )
8		csbeq1a 3542	. . . . . 6 |- ( x = z -> B = [_ z / x ]_ B )
9	8	sbceq1d 3440	. . . . 5 |- ( x = z -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
10	9	adantl 482	. . . 4 |- ( ( A. y F/ x ph /\ x = z ) -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
11		nfnf1 2031	. . . . 5 |- F/ x F/ x ph
12	11	nfal 2153	. . . 4 |- F/ x A. y F/ x ph
13		nfa1 2028	. . . . 5 |- F/ y A. y F/ x ph
14		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ z / x ]_ B
15	14	a1i 11	. . . . 5 |- ( A. y F/ x ph -> F/_ x [_ z / x ]_ B )
16		sp 2053	. . . . 5 |- ( A. y F/ x ph -> F/ x ph )
17	13, 15, 16	nfsbcd 3456	. . . 4 |- ( A. y F/ x ph -> F/ x [. [_ z / x ]_ B / y ]. ph )
18	7, 10, 12, 17	sbciedf 3471	. . 3 |- ( A. y F/ x ph -> ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
19	5, 18	vtoclg 3266	. 2 |- ( A e. V -> ( A. y F/ x ph -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) )
20	19	imp 445	1 |- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbnestgf  3996  sbcnestg  3997