Theorem xnegeqi 45792

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 13134	. 2 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
3	1, 2	ax-mp 5	1 ⊢ -𝑒𝐴 = -𝑒𝐵

Syntax hints:   = wceq 1542  -_𝑒cxne 13035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-neg 11379  df-xneg 13038

This theorem is referenced by:  supminfxr2  45821  liminfvalxr  46135  liminf0  46145  liminfpnfuz  46168