Theorem xnegeq 10035

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 2236	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2236	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 8350	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3627	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3627	. 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 9980	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 9980	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2287	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1395   ifcif 3602   +oocpnf 8189   -oocmnf 8190   -ucneg 8329   -ecxne 9977

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-if 3603  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6010  df-neg 8331  df-xneg 9980

This theorem is referenced by:  xnegcl  10040  xnegneg  10041  xneg11  10042  xltnegi  10043  xnegid  10067  xnegdi  10076  xsubge0  10089  xposdif  10090  xlesubadd  10091  xrnegiso  11789  infxrnegsupex  11790  xrminmax  11792  xrminrecl  11800  xrminadd  11802  xblss2ps  15094  xblss2  15095