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Theorem xnegeq 9578
Description: Equality of two extended numbers with  -e in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegeq  |-  ( A  =  B  ->  -e
A  =  -e
B )

Proof of Theorem xnegeq
StepHypRef Expression
1 eqeq1 2124 . . 3  |-  ( A  =  B  ->  ( A  = +oo  <->  B  = +oo ) )
2 eqeq1 2124 . . . 4  |-  ( A  =  B  ->  ( A  = -oo  <->  B  = -oo ) )
3 negeq 7923 . . . 4  |-  ( A  =  B  ->  -u A  =  -u B )
42, 3ifbieq2d 3466 . . 3  |-  ( A  =  B  ->  if ( A  = -oo , +oo ,  -u A
)  =  if ( B  = -oo , +oo ,  -u B ) )
51, 4ifbieq2d 3466 . 2  |-  ( A  =  B  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  if ( B  = +oo , -oo ,  if ( B  = -oo , +oo ,  -u B ) ) )
6 df-xneg 9527 . 2  |-  -e
A  =  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )
7 df-xneg 9527 . 2  |-  -e
B  =  if ( B  = +oo , -oo ,  if ( B  = -oo , +oo ,  -u B ) )
85, 6, 73eqtr4g 2175 1  |-  ( A  =  B  ->  -e
A  =  -e
B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   ifcif 3444   +oocpnf 7765   -oocmnf 7766   -ucneg 7902    -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-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-if 3445  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745  df-neg 7904  df-xneg 9527
This theorem is referenced by:  xnegcl  9583  xnegneg  9584  xneg11  9585  xltnegi  9586  xnegid  9610  xnegdi  9619  xsubge0  9632  xposdif  9633  xlesubadd  9634  xrnegiso  10999  infxrnegsupex  11000  xrminmax  11002  xrminrecl  11010  xrminadd  11012  xblss2ps  12500  xblss2  12501
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