Theorem xnegeq 10052

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 → -𝑒𝐴 = -𝑒𝐵)

Proof of Theorem xnegeq

Step	Hyp	Ref	Expression
1		eqeq1 2236	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞))
2		eqeq1 2236	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞))
3		negeq 8362	. . . 4 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵)
4	2, 3	ifbieq2d 3628	. . 3 ⊢ (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵))
5	1, 4	ifbieq2d 3628	. 2 ⊢ (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)))
6		df-xneg 9997	. 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
7		df-xneg 9997	. 2 ⊢ -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))
8	5, 6, 7	3eqtr4g 2287	1 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)

Syntax hints:   → wi 4   = wceq 1395  ifcif 3603  +∞cpnf 8201  -∞cmnf 8202  -cneg 8341  -_𝑒cxne 9994

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2802  df-un 3202  df-if 3604  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-iota 5284  df-fv 5332  df-ov 6016  df-neg 8343  df-xneg 9997

This theorem is referenced by:  xnegcl  10057  xnegneg  10058  xneg11  10059  xltnegi  10060  xnegid  10084  xnegdi  10093  xsubge0  10106  xposdif  10107  xlesubadd  10108  xrnegiso  11813  infxrnegsupex  11814  xrminmax  11816  xrminrecl  11824  xrminadd  11826  xblss2ps  15118  xblss2  15119