Theorem cdleme31snd 35674

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Apr-2013.)

Hypotheses

Ref	Expression
cdleme31snd.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme31snd.n	|- N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) )
cdleme31snd.e	|- E = ( ( O .\/ U ) ./\ ( Q .\/ ( ( P .\/ O ) ./\ W ) ) )
cdleme31snd.o	|- O = ( ( S .\/ V ) ./\ ( P .\/ ( ( Q .\/ S ) ./\ W ) ) )

Assertion

Ref	Expression
cdleme31snd	|- ( S e. A -> [_ S / v ]_ [_ N / t ]_ D = E )

Distinct variable groups:   v, A   v, D   v, t, .\/   t, ./\ , v   t, O   t, P, v   t, Q, v   v, S   t, U, v   v, V   t, W, v

Allowed substitution hints:   A( t)   D( t)   S( t)   E( v, t)   N( v, t)   O( v)   V( t)

Proof of Theorem cdleme31snd

Step	Hyp	Ref	Expression
1		csbnestg 3998	. 2 |- ( S e. A -> [_ S / v ]_ [_ N / t ]_ D = [_ [_ S / v ]_ N / t ]_ D )
2		cdleme31snd.n	. . . . 5 |- N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) )
3		cdleme31snd.o	. . . . 5 |- O = ( ( S .\/ V ) ./\ ( P .\/ ( ( Q .\/ S ) ./\ W ) ) )
4	2, 3	cdleme31sc 35672	. . . 4 |- ( S e. A -> [_ S / v ]_ N = O )
5	4	csbeq1d 3540	. . 3 |- ( S e. A -> [_ [_ S / v ]_ N / t ]_ D = [_ O / t ]_ D )
6	3	ovexi 6679	. . . 4 |- O e. _V
7		cdleme31snd.d	. . . . 5 |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
8		cdleme31snd.e	. . . . 5 |- E = ( ( O .\/ U ) ./\ ( Q .\/ ( ( P .\/ O ) ./\ W ) ) )
9	7, 8	cdleme31sc 35672	. . . 4 |- ( O e. _V -> [_ O / t ]_ D = E )
10	6, 9	ax-mp 5	. . 3 |- [_ O / t ]_ D = E
11	5, 10	syl6eq 2672	. 2 |- ( S e. A -> [_ [_ S / v ]_ N / t ]_ D = E )
12	1, 11	eqtrd 2656	1 |- ( S e. A -> [_ S / v ]_ [_ N / t ]_ D = E )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_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  ax-nul 4789

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-eu 2474  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:  cdlemeg46ngfr  35806