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Theorem xnegeq 9663
 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
StepHypRef Expression
1 eqeq1 2147 . . 3 (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞))
2 eqeq1 2147 . . . 4 (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞))
3 negeq 8002 . . . 4 (𝐴 = 𝐵 → -𝐴 = -𝐵)
42, 3ifbieq2d 3502 . . 3 (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵))
51, 4ifbieq2d 3502 . 2 (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)))
6 df-xneg 9612 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
7 df-xneg 9612 . 2 -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))
85, 6, 73eqtr4g 2198 1 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332  ifcif 3480  +∞cpnf 7844  -∞cmnf 7845  -cneg 7981  -𝑒cxne 9609 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-rab 2426  df-v 2692  df-un 3081  df-if 3481  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-iota 5098  df-fv 5141  df-ov 5787  df-neg 7983  df-xneg 9612 This theorem is referenced by:  xnegcl  9668  xnegneg  9669  xneg11  9670  xltnegi  9671  xnegid  9695  xnegdi  9704  xsubge0  9717  xposdif  9718  xlesubadd  9719  xrnegiso  11086  infxrnegsupex  11087  xrminmax  11089  xrminrecl  11097  xrminadd  11099  xblss2ps  12635  xblss2  12636
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