Theorem xnegeqd 42977

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 12941	. 2 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → -𝑒𝐴 = -𝑒𝐵)

Syntax hints:   → wi 4   = wceq 1539  -_𝑒cxne 12845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-neg 11208  df-xneg 12848

This theorem is referenced by:  supminfxr  43004  supminfxr2  43009  supminfxrrnmpt  43011  monoord2xrv  43024  liminfvalxr  43324  liminfvalxrmpt  43327  liminfval4  43330  liminfval3  43331  limsupval4  43335  liminfvaluz2  43336  limsupvaluz4  43341  climliminflimsupd  43342  xlimpnfxnegmnf  43355  liminfpnfuz  43357  xlimpnfxnegmnf2  43399  smfliminflem  44363