Theorem csbif 3942

Description: Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)

Assertion

Ref	Expression
csbif	|- [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C )

Proof of Theorem csbif

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3377	. . . 4 |- ( y = A -> [_ y / x ]_ if ( ph , B , C ) = [_ A / x ]_ if ( ph , B , C ) )
2		dfsbcq2 3281	. . . . 5 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		csbeq1 3377	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 3377	. . . . 5 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	ifbieq12d 3919	. . . 4 |- ( y = A -> if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) )
6	1, 5	eqeq12d 2476	. . 3 |- ( y = A -> ( [_ y / x ]_ if ( ph , B , C ) = if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C ) <-> [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) ) )
7		vex 3059	. . . 4 |- y e. _V
8		nfs1v 2276	. . . . 5 |- F/ x [ y / x ] ph
9		nfcsb1v 3390	. . . . 5 |- F/_ x [_ y / x ]_ B
10		nfcsb1v 3390	. . . . 5 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfif 3921	. . . 4 |- F/_ x if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C )
12		sbequ12 2093	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
13		csbeq1a 3383	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 3383	. . . . 5 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	ifbieq12d 3919	. . . 4 |- ( x = y -> if ( ph , B , C ) = if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C ) )
16	7, 11, 15	csbief 3399	. . 3 |- [_ y / x ]_ if ( ph , B , C ) = if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C )
17	6, 16	vtoclg 3118	. 2 |- ( A e. _V -> [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) )
18		csbprc 3781	. . 3 |- ( -. A e. _V -> [_ A / x ]_ if ( ph , B , C ) = (/) )
19		csbprc 3781	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
20		csbprc 3781	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
21	19, 20	ifeq12d 3912	. . . 4 |- ( -. A e. _V -> if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) = if ( [. A / x ]. ph , (/) , (/) ) )
22		ifid 3929	. . . 4 |- if ( [. A / x ]. ph , (/) , (/) ) = (/)
23	21, 22	syl6req 2512	. . 3 |- ( -. A e. _V -> (/) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) )
24	18, 23	eqtrd 2495	. 2 |- ( -. A e. _V -> [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) )
25	17, 24	pm2.61i 169	1 |- [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C )

Syntax hints:   -. wn 3   = wceq 1454   [wsb 1807   e. wcel 1897   _Vcvv 3056   [.wsbc 3278   [_csb 3374   (/)c0 3742   ifcif 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893

This theorem is referenced by:  fvmptnn04if  19921  csbopg2  31769  csbrdgg  31774  csbfinxpg  31824  cdlemk40  34528