Theorem xnegeq 8740

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 2046	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2046	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 7204	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3352	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3352	. 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 8689	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 8689	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2097	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1243   ifcif 3331   +oocpnf 7057   -oocmnf 7058   -ucneg 7183   -ecxne 8686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-if 3332  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-neg 7185  df-xneg 8689

This theorem is referenced by:  xnegcl  8745  xnegneg  8746  xneg11  8747  xltnegi  8748