Theorem xnegeq 9350

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 2095	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2095	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 7736	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3419	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3419	. 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 9304	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 9304	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2146	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1290   ifcif 3397   +oocpnf 7580   -oocmnf 7581   -ucneg 7715   -ecxne 9301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-rab 2369  df-v 2622  df-un 3004  df-if 3398  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669  df-neg 7717  df-xneg 9304

This theorem is referenced by:  xnegcl  9355  xnegneg  9356  xneg11  9357  xltnegi  9358