Theorem xnegeq 9951

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 2212	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2212	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 8267	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3595	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3595	. 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 9896	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 9896	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2263	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1373   ifcif 3571   +oocpnf 8106   -oocmnf 8107   -ucneg 8246   -ecxne 9893

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-if 3572  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-iota 5233  df-fv 5280  df-ov 5949  df-neg 8248  df-xneg 9896

This theorem is referenced by:  xnegcl  9956  xnegneg  9957  xneg11  9958  xltnegi  9959  xnegid  9983  xnegdi  9992  xsubge0  10005  xposdif  10006  xlesubadd  10007  xrnegiso  11606  infxrnegsupex  11607  xrminmax  11609  xrminrecl  11617  xrminadd  11619  xblss2ps  14909  xblss2  14910