Theorem xnegeq 9896

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 2200	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2200	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 8214	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3582	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3582	. 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 9841	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 9841	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2251	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1364   ifcif 3558   +oocpnf 8053   -oocmnf 8054   -ucneg 8193   -ecxne 9838

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3158  df-if 3559  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922  df-neg 8195  df-xneg 9841

This theorem is referenced by:  xnegcl  9901  xnegneg  9902  xneg11  9903  xltnegi  9904  xnegid  9928  xnegdi  9937  xsubge0  9950  xposdif  9951  xlesubadd  9952  xrnegiso  11408  infxrnegsupex  11409  xrminmax  11411  xrminrecl  11419  xrminadd  11421  xblss2ps  14583  xblss2  14584