Theorem xnegeqd 42958

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 12951	. 2 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → -𝑒𝐴 = -𝑒𝐵)

Syntax hints:   → wi 4   = wceq 1539  -_𝑒cxne 12855

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 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-iota 6384  df-fv 6434  df-ov 7270  df-neg 11218  df-xneg 12858

This theorem is referenced by:  supminfxr  42985  supminfxr2  42990  supminfxrrnmpt  42992  monoord2xrv  43005  liminfvalxr  43305  liminfvalxrmpt  43308  liminfval4  43311  liminfval3  43312  limsupval4  43316  liminfvaluz2  43317  limsupvaluz4  43322  climliminflimsupd  43323  xlimpnfxnegmnf  43336  liminfpnfuz  43338  xlimpnfxnegmnf2  43380  smfliminflem  44341