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Theorem xrnegcon1d 11026
Description: Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)
Hypotheses
Ref Expression
xrnegcon1d.a  |-  ( ph  ->  A  e.  RR* )
xrnegcon1d.b  |-  ( ph  ->  B  e.  RR* )
Assertion
Ref Expression
xrnegcon1d  |-  ( ph  ->  (  -e A  =  B  <->  -e B  =  A ) )

Proof of Theorem xrnegcon1d
StepHypRef Expression
1 xrnegcon1d.b . . . 4  |-  ( ph  ->  B  e.  RR* )
2 xnegneg 9609 . . . . 5  |-  ( B  e.  RR*  ->  -e  -e B  =  B )
32eqeq2d 2149 . . . 4  |-  ( B  e.  RR*  ->  (  -e A  =  -e  -e B  <->  -e A  =  B ) )
41, 3syl 14 . . 3  |-  ( ph  ->  (  -e A  =  -e  -e B  <->  -e A  =  B ) )
5 xrnegcon1d.a . . . 4  |-  ( ph  ->  A  e.  RR* )
61xnegcld 9631 . . . 4  |-  ( ph  -> 
-e B  e. 
RR* )
7 xneg11 9610 . . . 4  |-  ( ( A  e.  RR*  /\  -e
B  e.  RR* )  ->  (  -e A  =  -e  -e B  <->  A  =  -e
B ) )
85, 6, 7syl2anc 408 . . 3  |-  ( ph  ->  (  -e A  =  -e  -e B  <->  A  =  -e
B ) )
94, 8bitr3d 189 . 2  |-  ( ph  ->  (  -e A  =  B  <->  A  =  -e B ) )
10 eqcom 2139 . 2  |-  ( A  =  -e B  <->  -e B  =  A )
119, 10syl6bb 195 1  |-  ( ph  ->  (  -e A  =  B  <->  -e B  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   RR*cxr 7792    -ecxne 9549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-sub 7928  df-neg 7929  df-xneg 9552
This theorem is referenced by:  xrminmax  11027  xrmineqinf  11031
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