Theorem cdleme31sdnN 35675

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme31sdn.c	|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme31sdn.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme31sdn.n	|- N = if ( s .<_ ( P .\/ Q ) , I , C )

Assertion

Ref	Expression
cdleme31sdnN	|- N = if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D )

Distinct variable groups:   t, .\/   t, ./\   t, P   t, Q   t, U   t, W   t, s

Allowed substitution hints:   C( t, s)   D( t, s)   P( s)   Q( s)   U( s)   I( t, s)   .\/ ( s)   .<_ ( t, s)   ./\ ( s)   N( t, s)   W( s)

Proof of Theorem cdleme31sdnN

Step	Hyp	Ref	Expression
1		cdleme31sdn.n	. 2 |- N = if ( s .<_ ( P .\/ Q ) , I , C )
2		biid 251	. . 3 |- ( s .<_ ( P .\/ Q ) <-> s .<_ ( P .\/ Q ) )
3		vex 3203	. . . 4 |- s e. _V
4		cdleme31sdn.d	. . . . 5 |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
5		cdleme31sdn.c	. . . . 5 |- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
6	4, 5	cdleme31sc 35672	. . . 4 |- ( s e. _V -> [_ s / t ]_ D = C )
7	3, 6	ax-mp 5	. . 3 |- [_ s / t ]_ D = C
8	2, 7	ifbieq2i 4110	. 2 |- if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) = if ( s .<_ ( P .\/ Q ) , I , C )
9	1, 8	eqtr4i 2647	1 |- N = if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D )

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_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-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: (None)