Theorem xnegeqi 46544

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 13341	. 2 ⊢ (𝐴 = 𝐵 → -e𝐴 = -e𝐵)
3	1, 2	ax-mp 5	1 ⊢ -e𝐴 = -e𝐵

This proof depends on syntax axioms:   = wceq 1569   -_ecxne 13242

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6547  df-ov 7424  df-neg 11548  df-xneg 13245

This theorem is used by:  supminfxr2  46573  liminfvalxr  46887  liminf0  46897  liminfpnfuz  46920