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Theorem xnegeq 8740
 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 2046 . . 3 (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞))
2 eqeq1 2046 . . . 4 (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞))
3 negeq 7204 . . . 4 (𝐴 = 𝐵 → -𝐴 = -𝐵)
42, 3ifbieq2d 3352 . . 3 (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵))
51, 4ifbieq2d 3352 . 2 (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)))
6 df-xneg 8689 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
7 df-xneg 8689 . 2 -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))
85, 6, 73eqtr4g 2097 1 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  ifcif 3331  +∞cpnf 7057  -∞cmnf 7058  -cneg 7183  -𝑒cxne 8686 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-if 3332  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-neg 7185  df-xneg 8689 This theorem is referenced by:  xnegcl  8745  xnegneg  8746  xneg11  8747  xltnegi  8748
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