Theorem csbif 3984

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 3433	. . . 4 |- ( y = A -> [_ y / x ]_ if ( ph , B , C ) = [_ A / x ]_ if ( ph , B , C ) )
2		dfsbcq2 3329	. . . . 5 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		csbeq1 3433	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 3433	. . . . 5 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	ifbieq12d 3961	. . . 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 2484	. . 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 3111	. . . 4 |- y e. _V
8		nfs1v 2159	. . . . 5 |- F/ x [ y / x ] ph
9		nfcsb1v 3446	. . . . 5 |- F/_ x [_ y / x ]_ B
10		nfcsb1v 3446	. . . . 5 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfif 3963	. . . 4 |- F/_ x if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C )
12		sbequ12 1956	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
13		csbeq1a 3439	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 3439	. . . . 5 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	ifbieq12d 3961	. . . 4 |- ( x = y -> if ( ph , B , C ) = if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C ) )
16	7, 11, 15	csbief 3455	. . 3 |- [_ y / x ]_ if ( ph , B , C ) = if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C )
17	6, 16	vtoclg 3166	. 2 |- ( A e. _V -> [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) )
18		csbprc 3816	. . 3 |- ( -. A e. _V -> [_ A / x ]_ if ( ph , B , C ) = (/) )
19		csbprc 3816	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
20		csbprc 3816	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
21	19, 20	ifeq12d 3954	. . . 4 |- ( -. A e. _V -> if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) = if ( [. A / x ]. ph , (/) , (/) ) )
22		ifid 3971	. . . 4 |- if ( [. A / x ]. ph , (/) , (/) ) = (/)
23	21, 22	syl6req 2520	. . 3 |- ( -. A e. _V -> (/) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) )
24	18, 23	eqtrd 2503	. 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 164	1 |- [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C )

Syntax hints:   -. wn 3   = wceq 1374   [wsb 1706   e. wcel 1762   _Vcvv 3108   [.wsbc 3326   [_csb 3430   (/)c0 3780   ifcif 3934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935

This theorem is referenced by:  fvmptnn04if  19112  cdlemk40  35590