Theorem xnegeqi 45978

Description: Equality of two extended numbers with -_𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
xnegeqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
xnegeqi	⊢ -𝑒𝐴 = -𝑒𝐵

Proof of Theorem xnegeqi

Step	Hyp	Ref	Expression
1		xnegeqi.1	. 2 ⊢ 𝐴 = 𝐵
2		xnegeq 13207	. 2 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
3	1, 2	ax-mp 5	1 ⊢ -𝑒𝐴 = -𝑒𝐵

Syntax hints:   = wceq 1559  -_𝑒cxne 13108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-ov 7395  df-neg 11414  df-xneg 13111

This theorem is referenced by:  supminfxr2  46007  liminfvalxr  46321  liminf0  46331  liminfpnfuz  46354