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Theorem xnegeqd 40180
 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 12243 . 2 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
31, 2syl 17 1 (𝜑 → -𝑒𝐴 = -𝑒𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631  -𝑒cxne 12148 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-neg 10471  df-xneg 12151 This theorem is referenced by:  supminfxr  40210  supminfxr2  40215  supminfxrrnmpt  40217  monoord2xrv  40230  liminfvalxr  40533  liminfvalxrmpt  40536  liminfval4  40539  liminfval3  40540  limsupval4  40544  liminfvaluz2  40545  limsupvaluz4  40550  climliminflimsupd  40551  smfliminflem  41556
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