Theorem xnegeqi 46665

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

Step	Hyp	Ref	Expression
1		xnegeqi.1	. 2 ⊢ 𝐴 = 𝐵
2		xnegeq 13367	. 2 ⊢ (𝐴 = 𝐵 → -e𝐴 = -e𝐵)
3	1, 2	ax-mp 5	1 ⊢ -e𝐴 = -e𝐵

This proof depends on syntax axioms:   = wceq 1570   -_ecxne 13268

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-neg 11574  df-xneg 13271

This theorem is used by:  supminfxr2  46694  liminfvalxr  47008  liminf0  47018  liminfpnfuz  47041