Theorem csbif 4138

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 3536	. . . 4 |- ( y = A -> [_ y / x ]_ if ( ph , B , C ) = [_ A / x ]_ if ( ph , B , C ) )
2		dfsbcq2 3438	. . . . 5 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		csbeq1 3536	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4		csbeq1 3536	. . . . 5 |- ( y = A -> [_ y / x ]_ C = [_ A / x ]_ C )
5	2, 3, 4	ifbieq12d 4113	. . . 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 2637	. . 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 3203	. . . 4 |- y e. _V
8		nfs1v 2437	. . . . 5 |- F/ x [ y / x ] ph
9		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ B
10		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ C
11	8, 9, 10	nfif 4115	. . . 4 |- F/_ x if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C )
12		sbequ12 2111	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
13		csbeq1a 3542	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
14		csbeq1a 3542	. . . . 5 |- ( x = y -> C = [_ y / x ]_ C )
15	12, 13, 14	ifbieq12d 4113	. . . 4 |- ( x = y -> if ( ph , B , C ) = if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C ) )
16	7, 11, 15	csbief 3558	. . 3 |- [_ y / x ]_ if ( ph , B , C ) = if ( [ y / x ] ph , [_ y / x ]_ B , [_ y / x ]_ C )
17	6, 16	vtoclg 3266	. 2 |- ( A e. _V -> [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) )
18		csbprc 3980	. . 3 |- ( -. A e. _V -> [_ A / x ]_ if ( ph , B , C ) = (/) )
19		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
20		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
21	19, 20	ifeq12d 4106	. . . 4 |- ( -. A e. _V -> if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) = if ( [. A / x ]. ph , (/) , (/) ) )
22		ifid 4125	. . . 4 |- if ( [. A / x ]. ph , (/) , (/) ) = (/)
23	21, 22	syl6req 2673	. . 3 |- ( -. A e. _V -> (/) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) )
24	18, 23	eqtrd 2656	. 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 176	1 |- [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C )

Syntax hints:   -. wn 3   = wceq 1483   [wsb 1880   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915   ifcif 4086

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-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-nul 3916  df-if 4087

This theorem is referenced by:  csbopg  4420  fvmptnn04if  20654  csbrdgg  33175  csbfinxpg  33225  cdlemk40  36205