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Theorem wloglei 9559
Description: Form of wlogle 9560 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 452 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  S  C_  RR )
3 simprr 734 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  S )
42, 3sseldd 3349 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  RR )
5 simprl 733 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
62, 5sseldd 3349 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  RR )
7 vex 2959 . . 3  |-  x  e. 
_V
8 vex 2959 . . 3  |-  y  e. 
_V
9 eleq1 2496 . . . . . . 7  |-  ( z  =  x  ->  (
z  e.  S  <->  x  e.  S ) )
10 eleq1 2496 . . . . . . 7  |-  ( w  =  y  ->  (
w  e.  S  <->  y  e.  S ) )
119, 10bi2anan9 844 . . . . . 6  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( z  e.  S  /\  w  e.  S )  <->  ( x  e.  S  /\  y  e.  S ) ) )
1211anbi2d 685 . . . . 5  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( ph  /\  ( z  e.  S  /\  w  e.  S
) )  <->  ( ph  /\  ( x  e.  S  /\  y  e.  S
) ) ) )
13 breq12 4217 . . . . . 6  |-  ( ( w  =  y  /\  z  =  x )  ->  ( w  <_  z  <->  y  <_  x ) )
1413ancoms 440 . . . . 5  |-  ( ( z  =  x  /\  w  =  y )  ->  ( w  <_  z  <->  y  <_  x ) )
1512, 14anbi12d 692 . . . 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 312 . . 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 2959 . . . 4  |-  z  e. 
_V
19 vex 2959 . . . 4  |-  w  e. 
_V
20 ancom 438 . . . . . . . 8  |-  ( ( x  e.  S  /\  y  e.  S )  <->  ( y  e.  S  /\  x  e.  S )
)
21 eleq1 2496 . . . . . . . . 9  |-  ( y  =  z  ->  (
y  e.  S  <->  z  e.  S ) )
22 eleq1 2496 . . . . . . . . 9  |-  ( x  =  w  ->  (
x  e.  S  <->  w  e.  S ) )
2321, 22bi2anan9 844 . . . . . . . 8  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( y  e.  S  /\  x  e.  S )  <->  ( z  e.  S  /\  w  e.  S ) ) )
2420, 23syl5bb 249 . . . . . . 7  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( x  e.  S  /\  y  e.  S )  <->  ( z  e.  S  /\  w  e.  S ) ) )
2524anbi2d 685 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( ph  /\  ( x  e.  S  /\  y  e.  S
) )  <->  ( ph  /\  ( z  e.  S  /\  w  e.  S
) ) ) )
26 breq12 4217 . . . . . . 7  |-  ( ( x  =  w  /\  y  =  z )  ->  ( x  <_  y  <->  w  <_  z ) )
2726ancoms 440 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( x  <_  y  <->  w  <_  z ) )
2825, 27anbi12d 692 . . . . 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 1692 . . . . . . 7  |-  ( y  =  z  <->  z  =  y )
30 equcom 1692 . . . . . . 7  |-  ( x  =  w  <->  w  =  x )
31 wlogle.2 . . . . . . 7  |-  ( ( z  =  y  /\  w  =  x )  ->  ( ps  <->  th )
)
3229, 30, 31syl2anb 466 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ps  <->  th )
)
3332bicomd 193 . . . . 5  |-  ( ( y  =  z  /\  x  =  w )  ->  ( th  <->  ps )
)
3428, 33imbi12d 312 . . . 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 938 . . . . . 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 463 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  S  /\  y  e.  S )  /\  x  <_  y ) )  ->  th )
3837anassrs 630 . . . 4  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  x  <_  y )  ->  th )
3918, 19, 34, 38vtocl2 3007 . . 3  |-  ( ( ( ph  /\  (
z  e.  S  /\  w  e.  S )
)  /\  w  <_  z )  ->  ps )
407, 8, 17, 39vtocl2 3007 . 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 463 . . 3  |-  ( (
ph  /\  ( (
x  e.  S  /\  y  e.  S )  /\  x  <_  y ) )  ->  ch )
4342anassrs 630 . 2  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  x  <_  y )  ->  ch )
444, 6, 40, 43lecasei 9179 1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725    C_ wss 3320   class class class wbr 4212   RRcr 8989    <_ cle 9121
This theorem is referenced by:  wlogle  9560  rescon  24933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-pre-lttri 9064
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126
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