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Theorem xnegeqd 41587
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 12588 . 2 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
31, 2syl 17 1 (𝜑 → -𝑒𝐴 = -𝑒𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  -𝑒cxne 12492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-neg 10861  df-xneg 12495
This theorem is referenced by:  supminfxr  41616  supminfxr2  41621  supminfxrrnmpt  41623  monoord2xrv  41636  liminfvalxr  41940  liminfvalxrmpt  41943  liminfval4  41946  liminfval3  41947  limsupval4  41951  liminfvaluz2  41952  limsupvaluz4  41957  climliminflimsupd  41958  xlimpnfxnegmnf  41971  liminfpnfuz  41973  xlimpnfxnegmnf2  42015  smfliminflem  42981
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