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Theorem xnegeqi 42518
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 12683 . 2 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
31, 2ax-mp 5 1 -𝑒𝐴 = -𝑒𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  -𝑒cxne 12587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-ov 7173  df-neg 10951  df-xneg 12590
This theorem is referenced by:  supminfxr2  42549  liminfvalxr  42866  liminf0  42876  liminfpnfuz  42899
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