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Theorem xrnegcon1d 11001
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 9584 . . . . 5  |-  ( B  e.  RR*  ->  -e  -e B  =  B )
32eqeq2d 2129 . . . 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 9606 . . . 4  |-  ( ph  -> 
-e B  e. 
RR* )
7 xneg11 9585 . . . 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 2119 . 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 1316    e. wcel 1465   RR*cxr 7767    -ecxne 9524
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-sub 7903  df-neg 7904  df-xneg 9527
This theorem is referenced by:  xrminmax  11002  xrmineqinf  11006
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