Theorem xnegeq 10246

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 2245	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2245	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 8526	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 3665	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 3665	. 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 10191	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 10191	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2296	1 |- ( A = B -> -e A = -e B )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   ifcif 3638   +oocpnf 8363   -oocmnf 8364   -ucneg 8505   -ecxne 10188

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-if 3639  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-iota 5340  df-fv 5388  df-ov 6091  df-neg 8507  df-xneg 10191

This theorem is used by:  xnegcl  10251  xnegneg  10252  xneg11  10253  xltnegi  10254  xnegid  10278  xnegdi  10287  xsubge0  10300  xposdif  10301  xlesubadd  10302  xrnegiso  12053  infxrnegsupex  12054  xrminmax  12056  xrminrecl  12064  xrminadd  12066  xblss2ps  15630  xblss2  15631