Theorem xnegeq 10160

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 2239	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞))
2		eqeq1 2239	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞))
3		negeq 8466	. . . 4 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵)
4	2, 3	ifbieq2d 3647	. . 3 ⊢ (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵))
5	1, 4	ifbieq2d 3647	. 2 ⊢ (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)))
6		df-xneg 10105	. 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
7		df-xneg 10105	. 2 ⊢ -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))
8	5, 6, 7	3eqtr4g 2290	1 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)

Syntax hints:   → wi 4   = wceq 1398  ifcif 3620  +∞cpnf 8305  -∞cmnf 8306  -cneg 8445  -_𝑒cxne 10102

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-if 3621  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053  df-neg 8447  df-xneg 10105

This theorem is referenced by:  xnegcl  10165  xnegneg  10166  xneg11  10167  xltnegi  10168  xnegid  10192  xnegdi  10201  xsubge0  10214  xposdif  10215  xlesubadd  10216  xrnegiso  11947  infxrnegsupex  11948  xrminmax  11950  xrminrecl  11958  xrminadd  11960  xblss2ps  15269  xblss2  15270