Theorem xnegeqd 46267

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 13261	. 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 13162

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7419  df-neg 11471  df-xneg 13165

This theorem is used by:  supminfxr  46294  supminfxr2  46299  supminfxrrnmpt  46301  monoord2xrv  46313  liminfvalxr  46613  liminfvalxrmpt  46616  liminfval4  46619  liminfval3  46620  limsupval4  46624  liminfvaluz2  46625  limsupvaluz4  46630  climliminflimsupd  46631  xlimpnfxnegmnf  46644  liminfpnfuz  46646  xlimpnfxnegmnf2  46688  smfliminflem  47660