Theorem xnegeq 9640

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

Step	Hyp	Ref	Expression
1		eqeq1 2147	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2147	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 7979	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3501	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3501	. 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 9589	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 9589	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2198	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1332   ifcif 3479   +oocpnf 7821   -oocmnf 7822   -ucneg 7958   -ecxne 9586

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-rab 2426  df-v 2691  df-un 3080  df-if 3480  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785  df-neg 7960  df-xneg 9589

This theorem is referenced by:  xnegcl  9645  xnegneg  9646  xneg11  9647  xltnegi  9648  xnegid  9672  xnegdi  9681  xsubge0  9694  xposdif  9695  xlesubadd  9696  xrnegiso  11063  infxrnegsupex  11064  xrminmax  11066  xrminrecl  11074  xrminadd  11076  xblss2ps  12612  xblss2  12613