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Theorem wloglei 9321
Description: Form of wlogle 9322 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 451 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  S  C_  RR )
3 simprr 733 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  S )
42, 3sseldd 3194 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  RR )
5 simprl 732 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
62, 5sseldd 3194 . 2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  RR )
7 vex 2804 . . 3  |-  x  e. 
_V
8 vex 2804 . . 3  |-  y  e. 
_V
9 eleq1 2356 . . . . . . 7  |-  ( z  =  x  ->  (
z  e.  S  <->  x  e.  S ) )
10 eleq1 2356 . . . . . . 7  |-  ( w  =  y  ->  (
w  e.  S  <->  y  e.  S ) )
119, 10bi2anan9 843 . . . . . 6  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( z  e.  S  /\  w  e.  S )  <->  ( x  e.  S  /\  y  e.  S ) ) )
1211anbi2d 684 . . . . 5  |-  ( ( z  =  x  /\  w  =  y )  ->  ( ( ph  /\  ( z  e.  S  /\  w  e.  S
) )  <->  ( ph  /\  ( x  e.  S  /\  y  e.  S
) ) ) )
13 breq12 4044 . . . . . 6  |-  ( ( w  =  y  /\  z  =  x )  ->  ( w  <_  z  <->  y  <_  x ) )
1413ancoms 439 . . . . 5  |-  ( ( z  =  x  /\  w  =  y )  ->  ( w  <_  z  <->  y  <_  x ) )
1512, 14anbi12d 691 . . . 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 311 . . 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 2804 . . . 4  |-  z  e. 
_V
19 vex 2804 . . . 4  |-  w  e. 
_V
20 ancom 437 . . . . . . . 8  |-  ( ( x  e.  S  /\  y  e.  S )  <->  ( y  e.  S  /\  x  e.  S )
)
21 eleq1 2356 . . . . . . . . 9  |-  ( y  =  z  ->  (
y  e.  S  <->  z  e.  S ) )
22 eleq1 2356 . . . . . . . . 9  |-  ( x  =  w  ->  (
x  e.  S  <->  w  e.  S ) )
2321, 22bi2anan9 843 . . . . . . . 8  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( y  e.  S  /\  x  e.  S )  <->  ( z  e.  S  /\  w  e.  S ) ) )
2420, 23syl5bb 248 . . . . . . 7  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( x  e.  S  /\  y  e.  S )  <->  ( z  e.  S  /\  w  e.  S ) ) )
2524anbi2d 684 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( ph  /\  ( x  e.  S  /\  y  e.  S
) )  <->  ( ph  /\  ( z  e.  S  /\  w  e.  S
) ) ) )
26 breq12 4044 . . . . . . 7  |-  ( ( x  =  w  /\  y  =  z )  ->  ( x  <_  y  <->  w  <_  z ) )
2726ancoms 439 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( x  <_  y  <->  w  <_  z ) )
2825, 27anbi12d 691 . . . . 5  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ( ( ph  /\  ( x  e.  S  /\  y  e.  S
) )  /\  x  <_  y )  <->  ( ( ph  /\  ( z  e.  S  /\  w  e.  S ) )  /\  w  <_  z ) ) )
29 eqcom 2298 . . . . . . 7  |-  ( y  =  z  <->  z  =  y )
30 eqcom 2298 . . . . . . 7  |-  ( x  =  w  <->  w  =  x )
31 wlogle.2 . . . . . . 7  |-  ( ( z  =  y  /\  w  =  x )  ->  ( ps  <->  th )
)
3229, 30, 31syl2anb 465 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  ( ps  <->  th )
)
3332bicomd 192 . . . . 5  |-  ( ( y  =  z  /\  x  =  w )  ->  ( th  <->  ps )
)
3428, 33imbi12d 311 . . . 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 936 . . . . . 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 462 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  S  /\  y  e.  S )  /\  x  <_  y ) )  ->  th )
3837anassrs 629 . . . 4  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  x  <_  y )  ->  th )
3918, 19, 34, 38vtocl2 2852 . . 3  |-  ( ( ( ph  /\  (
z  e.  S  /\  w  e.  S )
)  /\  w  <_  z )  ->  ps )
407, 8, 17, 39vtocl2 2852 . 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 462 . . 3  |-  ( (
ph  /\  ( (
x  e.  S  /\  y  e.  S )  /\  x  <_  y ) )  ->  ch )
4342anassrs 629 . 2  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  x  <_  y )  ->  ch )
444, 6, 40, 43lecasei 8942 1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   RRcr 8752    <_ cle 8884
This theorem is referenced by:  wlogle  9322  rescon  23792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-pre-lttri 8827
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889
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