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Theorem xnegeqi 40398
 Description: Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
xnegeqi.1 𝐴 = 𝐵
Assertion
Ref Expression
xnegeqi -𝑒𝐴 = -𝑒𝐵

Proof of Theorem xnegeqi
StepHypRef Expression
1 xnegeqi.1 . 2 𝐴 = 𝐵
2 xnegeq 12283 . 2 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
31, 2ax-mp 5 1 -𝑒𝐴 = -𝑒𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1653  -𝑒cxne 12186 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 2354  ax-ext 2775 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 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-iota 6062  df-fv 6107  df-ov 6879  df-neg 10557  df-xneg 12189 This theorem is referenced by:  supminfxr2  40430  liminfvalxr  40747  liminf0  40757
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