Theorem cdleme31se2 35671

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 3-Apr-2013.)

Hypotheses

Ref	Expression
cdleme31se2.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) )
cdleme31se2.y	|- Y = ( ( P .\/ Q ) ./\ ( [_ S / t ]_ D .\/ ( ( R .\/ S ) ./\ W ) ) )

Assertion

Ref	Expression
cdleme31se2	|- ( S e. A -> [_ S / t ]_ E = Y )

Distinct variable groups:   t, A   t, .\/   t, ./\   t, P   t, Q   t, R   t, S   t, W

Allowed substitution hints:   D( t)   E( t)   Y( t)

Proof of Theorem cdleme31se2

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . . 5 |- F/_ t ( P .\/ Q )
2		nfcv 2764	. . . . 5 |- F/_ t ./\
3		nfcsb1v 3549	. . . . . 6 |- F/_ t [_ S / t ]_ D
4		nfcv 2764	. . . . . 6 |- F/_ t .\/
5		nfcv 2764	. . . . . 6 |- F/_ t ( ( R .\/ S ) ./\ W )
6	3, 4, 5	nfov 6676	. . . . 5 |- F/_ t ( [_ S / t ]_ D .\/ ( ( R .\/ S ) ./\ W ) )
7	1, 2, 6	nfov 6676	. . . 4 |- F/_ t ( ( P .\/ Q ) ./\ ( [_ S / t ]_ D .\/ ( ( R .\/ S ) ./\ W ) ) )
8	7	a1i 11	. . 3 |- ( S e. A -> F/_ t ( ( P .\/ Q ) ./\ ( [_ S / t ]_ D .\/ ( ( R .\/ S ) ./\ W ) ) ) )
9		csbeq1a 3542	. . . . 5 |- ( t = S -> D = [_ S / t ]_ D )
10		oveq2 6658	. . . . . 6 |- ( t = S -> ( R .\/ t ) = ( R .\/ S ) )
11	10	oveq1d 6665	. . . . 5 |- ( t = S -> ( ( R .\/ t ) ./\ W ) = ( ( R .\/ S ) ./\ W ) )
12	9, 11	oveq12d 6668	. . . 4 |- ( t = S -> ( D .\/ ( ( R .\/ t ) ./\ W ) ) = ( [_ S / t ]_ D .\/ ( ( R .\/ S ) ./\ W ) ) )
13	12	oveq2d 6666	. . 3 |- ( t = S -> ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( [_ S / t ]_ D .\/ ( ( R .\/ S ) ./\ W ) ) ) )
14	8, 13	csbiegf 3557	. 2 |- ( S e. A -> [_ S / t ]_ ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( [_ S / t ]_ D .\/ ( ( R .\/ S ) ./\ W ) ) ) )
15		cdleme31se2.e	. . 3 |- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) )
16	15	csbeq2i 3993	. 2 |- [_ S / t ]_ E = [_ S / t ]_ ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) )
17		cdleme31se2.y	. 2 |- Y = ( ( P .\/ Q ) ./\ ( [_ S / t ]_ D .\/ ( ( R .\/ S ) ./\ W ) ) )
18	14, 16, 17	3eqtr4g 2681	1 |- ( S e. A -> [_ S / t ]_ E = Y )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   [_csb 3533  (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-ral 2917  df-rex 2918  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-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  cdlemeg47rv2  35798