Theorem xnegeq 10062

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 2238	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2238	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 8372	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3630	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3630	. 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 10007	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 10007	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2289	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1397   ifcif 3605   +oocpnf 8211   -oocmnf 8212   -ucneg 8351   -ecxne 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-if 3606  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021  df-neg 8353  df-xneg 10007

This theorem is referenced by:  xnegcl  10067  xnegneg  10068  xneg11  10069  xltnegi  10070  xnegid  10094  xnegdi  10103  xsubge0  10116  xposdif  10117  xlesubadd  10118  xrnegiso  11840  infxrnegsupex  11841  xrminmax  11843  xrminrecl  11851  xrminadd  11853  xblss2ps  15147  xblss2  15148