Theorem xnegeqd 46652

Description: Equality of two extended numbers with -_e in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
xnegeqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
xnegeqd	⊢ (𝜑 → -e𝐴 = -e𝐵)

Proof of Theorem xnegeqd

Step	Hyp	Ref	Expression
1		xnegeqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		xnegeq 13364	. 2 ⊢ (𝐴 = 𝐵 → -e𝐴 = -e𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → -e𝐴 = -e𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   -_ecxne 13265

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 6512  df-fv 6566  df-ov 7446  df-neg 11571  df-xneg 13268

This theorem is used by:  supminfxr  46679  supminfxr2  46684  supminfxrrnmpt  46686  monoord2xrv  46698  liminfvalxr  46998  liminfvalxrmpt  47001  liminfval4  47004  liminfval3  47005  limsupval4  47009  liminfvaluz2  47010  limsupvaluz4  47015  climliminflimsupd  47016  xlimpnfxnegmnf  47029  liminfpnfuz  47031  xlimpnfxnegmnf2  47073  smfliminflem  48045