Theorem xnegeq 10179

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 2241	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2241	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 8482	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3651	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3651	. 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 10124	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 10124	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2292	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1398   ifcif 3624   +oocpnf 8321   -oocmnf 8322   -ucneg 8461   -ecxne 10121

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-if 3625  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-neg 8463  df-xneg 10124

This theorem is referenced by:  xnegcl  10184  xnegneg  10185  xneg11  10186  xltnegi  10187  xnegid  10211  xnegdi  10220  xsubge0  10233  xposdif  10234  xlesubadd  10235  xrnegiso  11972  infxrnegsupex  11973  xrminmax  11975  xrminrecl  11983  xrminadd  11985  xblss2ps  15395  xblss2  15396