Theorem xnegeqd 46523

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 13338	. 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 13239

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-neg 11545  df-xneg 13242

This theorem is used by:  supminfxr  46550  supminfxr2  46555  supminfxrrnmpt  46557  monoord2xrv  46569  liminfvalxr  46869  liminfvalxrmpt  46872  liminfval4  46875  liminfval3  46876  limsupval4  46880  liminfvaluz2  46881  limsupvaluz4  46886  climliminflimsupd  46887  xlimpnfxnegmnf  46900  liminfpnfuz  46902  xlimpnfxnegmnf2  46944  smfliminflem  47916