Theorem xnegeqd 45784

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 13134	. 2 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → -𝑒𝐴 = -𝑒𝐵)

Syntax hints:   → wi 4   = 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:  supminfxr  45811  supminfxr2  45816  supminfxrrnmpt  45818  monoord2xrv  45830  liminfvalxr  46130  liminfvalxrmpt  46133  liminfval4  46136  liminfval3  46137  limsupval4  46141  liminfvaluz2  46142  limsupvaluz4  46147  climliminflimsupd  46148  xlimpnfxnegmnf  46161  liminfpnfuz  46163  xlimpnfxnegmnf2  46205  smfliminflem  47177