Theorem sbcnestgf 3796

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 3281	. . . . 5 |- ( z = A -> ( [. z / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / y ]. ph ) )
2		csbeq1 3378	. . . . . 6 |- ( z = A -> [_ z / x ]_ B = [_ A / x ]_ B )
3	2	sbceq1d 3284	. . . . 5 |- ( z = A -> ( [. [_ z / x ]_ B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
4	1, 3	bibi12d 327	. . . 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 322	. . 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 3060	. . . . 5 |- z e. _V
7	6	a1i 11	. . . 4 |- ( A. y F/ x ph -> z e. _V )
8		csbeq1a 3384	. . . . . 6 |- ( x = z -> B = [_ z / x ]_ B )
9	8	sbceq1d 3284	. . . . 5 |- ( x = z -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
10	9	adantl 472	. . . 4 |- ( ( A. y F/ x ph /\ x = z ) -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
11		nfnf1 1992	. . . . 5 |- F/ x F/ x ph
12	11	nfal 2041	. . . 4 |- F/ x A. y F/ x ph
13		nfa1 1990	. . . . 5 |- F/ y A. y F/ x ph
14		nfcsb1v 3391	. . . . . 6 |- F/_ x [_ z / x ]_ B
15	14	a1i 11	. . . . 5 |- ( A. y F/ x ph -> F/_ x [_ z / x ]_ B )
16		sp 1948	. . . . 5 |- ( A. y F/ x ph -> F/ x ph )
17	13, 15, 16	nfsbcd 3300	. . . 4 |- ( A. y F/ x ph -> F/ x [. [_ z / x ]_ B / y ]. ph )
18	7, 10, 12, 17	sbciedf 3315	. . 3 |- ( A. y F/ x ph -> ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
19	5, 18	vtoclg 3119	. 2 |- ( A e. V -> ( A. y F/ x ph -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) )
20	19	imp 435	1 |- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1453   = wceq 1455   F/wnf 1678   e. wcel 1898   F/_wnfc 2590   _Vcvv 3057   [.wsbc 3279   [_csb 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sbc 3280  df-csb 3376

This theorem is referenced by:  csbnestgf  3797  sbcnestg  3798