Theorem cdleme31sn 35668

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)

Hypotheses

Ref	Expression
cdleme31sn.n	|- N = if ( s .<_ ( P .\/ Q ) , I , D )
cdleme31sn.c	|- C = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D )

Assertion

Ref	Expression
cdleme31sn	|- ( R e. A -> [_ R / s ]_ N = C )

Distinct variable groups:   A, s   .\/ , s   .<_ , s   P, s   Q, s   R, s

Allowed substitution hints:   C( s)   D( s)   I( s)   N( s)

Proof of Theorem cdleme31sn

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . 5 |- F/ s R .<_ ( P .\/ Q )
2		nfcsb1v 3549	. . . . 5 |- F/_ s [_ R / s ]_ I
3		nfcsb1v 3549	. . . . 5 |- F/_ s [_ R / s ]_ D
4	1, 2, 3	nfif 4115	. . . 4 |- F/_ s if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D )
5	4	a1i 11	. . 3 |- ( R e. A -> F/_ s if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) )
6		breq1 4656	. . . 4 |- ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) )
7		csbeq1a 3542	. . . 4 |- ( s = R -> I = [_ R / s ]_ I )
8		csbeq1a 3542	. . . 4 |- ( s = R -> D = [_ R / s ]_ D )
9	6, 7, 8	ifbieq12d 4113	. . 3 |- ( s = R -> if ( s .<_ ( P .\/ Q ) , I , D ) = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) )
10	5, 9	csbiegf 3557	. 2 |- ( R e. A -> [_ R / s ]_ if ( s .<_ ( P .\/ Q ) , I , D ) = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) )
11		cdleme31sn.n	. . 3 |- N = if ( s .<_ ( P .\/ Q ) , I , D )
12	11	csbeq2i 3993	. 2 |- [_ R / s ]_ N = [_ R / s ]_ if ( s .<_ ( P .\/ Q ) , I , D )
13		cdleme31sn.c	. 2 |- C = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D )
14	10, 12, 13	3eqtr4g 2681	1 |- ( R e. A -> [_ R / s ]_ N = C )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   [_csb 3533   ifcif 4086   class class class wbr 4653  (class class class)co 6650

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-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-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  cdleme31sn1  35669  cdleme31sn2  35677