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Theorem xnegeqd 40402
Description: Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
xnegeqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
xnegeqd (𝜑 → -𝑒𝐴 = -𝑒𝐵)

Proof of Theorem xnegeqd
StepHypRef Expression
1 xnegeqd.1 . 2 (𝜑𝐴 = 𝐵)
2 xnegeq 12286 . 2 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
31, 2syl 17 1 (𝜑 → -𝑒𝐴 = -𝑒𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  -𝑒cxne 12189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-iota 6065  df-fv 6110  df-ov 6882  df-neg 10560  df-xneg 12192
This theorem is referenced by:  supminfxr  40432  supminfxr2  40437  supminfxrrnmpt  40439  monoord2xrv  40452  liminfvalxr  40754  liminfvalxrmpt  40757  liminfval4  40760  liminfval3  40761  limsupval4  40765  liminfvaluz2  40766  limsupvaluz4  40771  climliminflimsupd  40772  smfliminflem  41777
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