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Theorem wloglei 9877
Description: Form of wlogle 9878 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 465 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  S  C_  RR )
3 simprr 756 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  S )
42, 3sseldd 3362 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  RR )
5 simprl 755 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
62, 5sseldd 3362 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  RR )
7 vex 2980 . . 3  |-  x  e. 
_V
8 vex 2980 . . 3  |-  y  e. 
_V
9 eleq1 2503 . . . . . . 7  |-  ( z  =  x  ->  (
z  e.  S  <->  x  e.  S ) )
10 eleq1 2503 . . . . . . 7  |-  ( w  =  y  ->  (
w  e.  S  <->  y  e.  S ) )
119, 10bi2anan9 868 . . . . . 6  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( z  e.  S  /\  w  e.  S )  <->  ( x  e.  S  /\  y  e.  S ) ) )
1211anbi2d 703 . . . . 5  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( ph  /\  ( z  e.  S  /\  w  e.  S
) )  <->  ( ph  /\  ( x  e.  S  /\  y  e.  S
) ) ) )
13 breq12 4302 . . . . . 6  |-  ( ( w  =  y  /\  z  =  x )  ->  ( w  <_  z  <->  y  <_  x ) )
1413ancoms 453 . . . . 5  |-  ( ( z  =  x  /\  w  =  y )  ->  ( w  <_  z  <->  y  <_  x ) )
1512, 14anbi12d 710 . . . 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 320 . . 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 2980 . . . 4  |-  z  e. 
_V
19 vex 2980 . . . 4  |-  w  e. 
_V
20 ancom 450 . . . . . . . 8  |-  ( ( x  e.  S  /\  y  e.  S )  <->  ( y  e.  S  /\  x  e.  S )
)
21 eleq1 2503 . . . . . . . . 9  |-  ( y  =  z  ->  (
y  e.  S  <->  z  e.  S ) )
22 eleq1 2503 . . . . . . . . 9  |-  ( x  =  w  ->  (
x  e.  S  <->  w  e.  S ) )
2321, 22bi2anan9 868 . . . . . . . 8  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( y  e.  S  /\  x  e.  S )  <->  ( z  e.  S  /\  w  e.  S ) ) )
2420, 23syl5bb 257 . . . . . . 7  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( x  e.  S  /\  y  e.  S )  <->  ( z  e.  S  /\  w  e.  S ) ) )
2524anbi2d 703 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( ph  /\  ( x  e.  S  /\  y  e.  S
) )  <->  ( ph  /\  ( z  e.  S  /\  w  e.  S
) ) ) )
26 breq12 4302 . . . . . . 7  |-  ( ( x  =  w  /\  y  =  z )  ->  ( x  <_  y  <->  w  <_  z ) )
2726ancoms 453 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( x  <_  y  <->  w  <_  z ) )
2825, 27anbi12d 710 . . . . 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 1732 . . . . . . 7  |-  ( y  =  z  <->  z  =  y )
30 equcom 1732 . . . . . . 7  |-  ( x  =  w  <->  w  =  x )
31 wlogle.2 . . . . . . 7  |-  ( ( z  =  y  /\  w  =  x )  ->  ( ps  <->  th )
)
3229, 30, 31syl2anb 479 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ps  <->  th )
)
3332bicomd 201 . . . . 5  |-  ( ( y  =  z  /\  x  =  w )  ->  ( th  <->  ps )
)
3428, 33imbi12d 320 . . . 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 967 . . . . . 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 476 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  S  /\  y  e.  S )  /\  x  <_  y ) )  ->  th )
3837anassrs 648 . . . 4  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  x  <_  y )  ->  th )
3918, 19, 34, 38vtocl2 3030 . . 3  |-  ( ( ( ph  /\  (
z  e.  S  /\  w  e.  S )
)  /\  w  <_  z )  ->  ps )
407, 8, 17, 39vtocl2 3030 . 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 476 . . 3  |-  ( (
ph  /\  ( (
x  e.  S  /\  y  e.  S )  /\  x  <_  y ) )  ->  ch )
4342anassrs 648 . 2  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  x  <_  y )  ->  ch )
444, 6, 40, 43lecasei 9485 1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1756    C_ wss 3333   class class class wbr 4297   RRcr 9286    <_ cle 9424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-pre-lttri 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429
This theorem is referenced by:  wlogle  9878  rescon  27140
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