Theorem xnegeqi 45355

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 13269	. 2 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
3	1, 2	ax-mp 5	1 ⊢ -𝑒𝐴 = -𝑒𝐵

Syntax hints:   = wceq 1537  -_𝑒cxne 13172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-neg 11523  df-xneg 13175

This theorem is referenced by:  supminfxr2  45384  liminfvalxr  45704  liminf0  45714  liminfpnfuz  45737