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Theorem wloglei 10146
Description: Form of wlogle 10147 where both sides of the equivalence are proven rather than showing that they are equivalent to each other. (Contributed by Mario Carneiro, 9-Mar-2015.)
Hypotheses
Ref Expression
wlogle.1  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ps  <->  ch )
)
wlogle.2  |-  ( ( z  =  y  /\  w  =  x )  ->  ( ps  <->  th )
)
wlogle.3  |-  ( ph  ->  S  C_  RR )
wloglei.4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  th )
wloglei.5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  ch )
Assertion
Ref Expression
wloglei  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ch )
Distinct variable groups:    x, w, y, z, ph    w, S, x, y, z    ps, x, y    ch, w, z
Allowed substitution hints:    ps( z, w)    ch( x, y)    th( x, y, z, w)

Proof of Theorem wloglei
StepHypRef Expression
1 wlogle.3 . . . 4  |-  ( ph  ->  S  C_  RR )
21adantr 467 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  S  C_  RR )
3 simprr 766 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  S )
42, 3sseldd 3433 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  RR )
5 simprl 764 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
62, 5sseldd 3433 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  RR )
7 vex 3048 . . 3  |-  x  e. 
_V
8 vex 3048 . . 3  |-  y  e. 
_V
9 eleq1 2517 . . . . . . 7  |-  ( z  =  x  ->  (
z  e.  S  <->  x  e.  S ) )
10 eleq1 2517 . . . . . . 7  |-  ( w  =  y  ->  (
w  e.  S  <->  y  e.  S ) )
119, 10bi2anan9 884 . . . . . 6  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( z  e.  S  /\  w  e.  S )  <->  ( x  e.  S  /\  y  e.  S ) ) )
1211anbi2d 710 . . . . 5  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( ph  /\  ( z  e.  S  /\  w  e.  S
) )  <->  ( ph  /\  ( x  e.  S  /\  y  e.  S
) ) ) )
13 breq12 4407 . . . . . 6  |-  ( ( w  =  y  /\  z  =  x )  ->  ( w  <_  z  <->  y  <_  x ) )
1413ancoms 455 . . . . 5  |-  ( ( z  =  x  /\  w  =  y )  ->  ( w  <_  z  <->  y  <_  x ) )
1512, 14anbi12d 717 . . . 4  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( ( ph  /\  ( z  e.  S  /\  w  e.  S
) )  /\  w  <_  z )  <->  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  /\  y  <_  x ) ) )
16 wlogle.1 . . . 4  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ps  <->  ch )
)
1715, 16imbi12d 322 . . 3  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( ( (
ph  /\  ( z  e.  S  /\  w  e.  S ) )  /\  w  <_  z )  ->  ps )  <->  ( ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  /\  y  <_  x )  ->  ch ) ) )
18 vex 3048 . . . 4  |-  z  e. 
_V
19 vex 3048 . . . 4  |-  w  e. 
_V
20 ancom 452 . . . . . . . 8  |-  ( ( x  e.  S  /\  y  e.  S )  <->  ( y  e.  S  /\  x  e.  S )
)
21 eleq1 2517 . . . . . . . . 9  |-  ( y  =  z  ->  (
y  e.  S  <->  z  e.  S ) )
22 eleq1 2517 . . . . . . . . 9  |-  ( x  =  w  ->  (
x  e.  S  <->  w  e.  S ) )
2321, 22bi2anan9 884 . . . . . . . 8  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( y  e.  S  /\  x  e.  S )  <->  ( z  e.  S  /\  w  e.  S ) ) )
2420, 23syl5bb 261 . . . . . . 7  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( x  e.  S  /\  y  e.  S )  <->  ( z  e.  S  /\  w  e.  S ) ) )
2524anbi2d 710 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( ph  /\  ( x  e.  S  /\  y  e.  S
) )  <->  ( ph  /\  ( z  e.  S  /\  w  e.  S
) ) ) )
26 breq12 4407 . . . . . . 7  |-  ( ( x  =  w  /\  y  =  z )  ->  ( x  <_  y  <->  w  <_  z ) )
2726ancoms 455 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( x  <_  y  <->  w  <_  z ) )
2825, 27anbi12d 717 . . . . 5  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( ( ph  /\  ( x  e.  S  /\  y  e.  S
) )  /\  x  <_  y )  <->  ( ( ph  /\  ( z  e.  S  /\  w  e.  S ) )  /\  w  <_  z ) ) )
29 equcom 1862 . . . . . . 7  |-  ( y  =  z  <->  z  =  y )
30 equcom 1862 . . . . . . 7  |-  ( x  =  w  <->  w  =  x )
31 wlogle.2 . . . . . . 7  |-  ( ( z  =  y  /\  w  =  x )  ->  ( ps  <->  th )
)
3229, 30, 31syl2anb 482 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ps  <->  th )
)
3332bicomd 205 . . . . 5  |-  ( ( y  =  z  /\  x  =  w )  ->  ( th  <->  ps )
)
3428, 33imbi12d 322 . . . 4  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  /\  x  <_  y )  ->  th )  <->  ( ( (
ph  /\  ( z  e.  S  /\  w  e.  S ) )  /\  w  <_  z )  ->  ps ) ) )
35 df-3an 987 . . . . . 6  |-  ( ( x  e.  S  /\  y  e.  S  /\  x  <_  y )  <->  ( (
x  e.  S  /\  y  e.  S )  /\  x  <_  y ) )
36 wloglei.4 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  th )
3735, 36sylan2br 479 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  S  /\  y  e.  S )  /\  x  <_  y ) )  ->  th )
3837anassrs 654 . . . 4  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  x  <_  y )  ->  th )
3918, 19, 34, 38vtocl2 3102 . . 3  |-  ( ( ( ph  /\  (
z  e.  S  /\  w  e.  S )
)  /\  w  <_  z )  ->  ps )
407, 8, 17, 39vtocl2 3102 . 2  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  y  <_  x )  ->  ch )
41 wloglei.5 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  x  <_  y ) )  ->  ch )
4235, 41sylan2br 479 . . 3  |-  ( (
ph  /\  ( (
x  e.  S  /\  y  e.  S )  /\  x  <_  y ) )  ->  ch )
4342anassrs 654 . 2  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  x  <_  y )  ->  ch )
444, 6, 40, 43lecasei 9740 1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    e. wcel 1887    C_ wss 3404   class class class wbr 4402   RRcr 9538    <_ cle 9676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-resscn 9596  ax-pre-lttri 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681
This theorem is referenced by:  wlogle  10147  rescon  29969
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