Theorem xnegeqd 46369

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 13307	. 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 13208

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 2732

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-ov 7411  df-neg 11516  df-xneg 13211

This theorem is used by:  supminfxr  46396  supminfxr2  46401  supminfxrrnmpt  46403  monoord2xrv  46415  liminfvalxr  46715  liminfvalxrmpt  46718  liminfval4  46721  liminfval3  46722  limsupval4  46726  liminfvaluz2  46727  limsupvaluz4  46732  climliminflimsupd  46733  xlimpnfxnegmnf  46746  liminfpnfuz  46748  xlimpnfxnegmnf2  46790  smfliminflem  47762