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Theorem xnegeqi 42980
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 12941 . 2 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
31, 2ax-mp 5 1 -𝑒𝐴 = -𝑒𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  -𝑒cxne 12845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-neg 11208  df-xneg 12848
This theorem is referenced by:  supminfxr2  43009  liminfvalxr  43324  liminf0  43334  liminfpnfuz  43357
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