Theorem sbcnestgf 3814

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 3301	. . . . 5 |- ( z = A -> ( [. z / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / y ]. ph ) )
2		csbeq1 3398	. . . . . 6 |- ( z = A -> [_ z / x ]_ B = [_ A / x ]_ B )
3	2	sbceq1d 3304	. . . . 5 |- ( z = A -> ( [. [_ z / x ]_ B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
4	1, 3	bibi12d 322	. . . 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 317	. . 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 3083	. . . . 5 |- z e. _V
7	6	a1i 11	. . . 4 |- ( A. y F/ x ph -> z e. _V )
8		csbeq1a 3404	. . . . . 6 |- ( x = z -> B = [_ z / x ]_ B )
9	8	sbceq1d 3304	. . . . 5 |- ( x = z -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
10	9	adantl 467	. . . 4 |- ( ( A. y F/ x ph /\ x = z ) -> ( [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
11		nfnf1 1958	. . . . 5 |- F/ x F/ x ph
12	11	nfal 2007	. . . 4 |- F/ x A. y F/ x ph
13		nfa1 1956	. . . . 5 |- F/ y A. y F/ x ph
14		nfcsb1v 3411	. . . . . 6 |- F/_ x [_ z / x ]_ B
15	14	a1i 11	. . . . 5 |- ( A. y F/ x ph -> F/_ x [_ z / x ]_ B )
16		sp 1914	. . . . 5 |- ( A. y F/ x ph -> F/ x ph )
17	13, 15, 16	nfsbcd 3320	. . . 4 |- ( A. y F/ x ph -> F/ x [. [_ z / x ]_ B / y ]. ph )
18	7, 10, 12, 17	sbciedf 3335	. . 3 |- ( A. y F/ x ph -> ( [. z / x ]. [. B / y ]. ph <-> [. [_ z / x ]_ B / y ]. ph ) )
19	5, 18	vtoclg 3139	. 2 |- ( A e. V -> ( A. y F/ x ph -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) )
20	19	imp 430	1 |- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435   = wceq 1437   F/wnf 1661   e. wcel 1872   F/_wnfc 2566   _Vcvv 3080   [.wsbc 3299   [_csb 3395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401

This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-sbc 3300  df-csb 3396

This theorem is referenced by:  csbnestgf  3815  sbcnestg  3816