Theorem xnegeqd 46179

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 13239	. 2 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → -𝑒𝐴 = -𝑒𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  -_𝑒cxne 13140

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415  df-neg 11450  df-xneg 13143

This theorem is used by:  supminfxr  46206  supminfxr2  46211  supminfxrrnmpt  46213  monoord2xrv  46225  liminfvalxr  46525  liminfvalxrmpt  46528  liminfval4  46531  liminfval3  46532  limsupval4  46536  liminfvaluz2  46537  limsupvaluz4  46542  climliminflimsupd  46543  xlimpnfxnegmnf  46556  liminfpnfuz  46558  xlimpnfxnegmnf2  46600  smfliminflem  47572