Theorem xnegeq 9823

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 2184	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2184	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 8146	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3558	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3558	. 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 9768	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 9768	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2235	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1353   ifcif 3534   +oocpnf 7985   -oocmnf 7986   -ucneg 8125   -ecxne 9765

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-if 3535  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-ov 5875  df-neg 8127  df-xneg 9768

This theorem is referenced by:  xnegcl  9828  xnegneg  9829  xneg11  9830  xltnegi  9831  xnegid  9855  xnegdi  9864  xsubge0  9877  xposdif  9878  xlesubadd  9879  xrnegiso  11263  infxrnegsupex  11264  xrminmax  11266  xrminrecl  11274  xrminadd  11276  xblss2ps  13775  xblss2  13776