Theorem xnegeq 9603

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 2144	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2144	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 7948	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3491	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3491	. 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 9552	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 9552	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2195	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1331   ifcif 3469   +oocpnf 7790   -oocmnf 7791   -ucneg 7927   -ecxne 9549

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-rab 2423  df-v 2683  df-un 3070  df-if 3470  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770  df-neg 7929  df-xneg 9552

This theorem is referenced by:  xnegcl  9608  xnegneg  9609  xneg11  9610  xltnegi  9611  xnegid  9635  xnegdi  9644  xsubge0  9657  xposdif  9658  xlesubadd  9659  xrnegiso  11024  infxrnegsupex  11025  xrminmax  11027  xrminrecl  11035  xrminadd  11037  xblss2ps  12562  xblss2  12563