Theorem nfxneg 45647

Description: Bound-variable hypothesis builder for the negative of an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
nfxneg.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfxneg	⊢ Ⅎ𝑥-𝑒𝐴

Proof of Theorem nfxneg

Step	Hyp	Ref	Expression
1		nfxneg.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
3	2	nfxnegd 45627	. 2 ⊢ (⊤ → Ⅎ𝑥-𝑒𝐴)
4	3	mptru 1548	1 ⊢ Ⅎ𝑥-𝑒𝐴

Syntax hints:  ⊤wtru 1542  Ⅎwnfc 2881  -_𝑒cxne 13021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359  df-neg 11365  df-xneg 13024

This theorem is referenced by:  liminfvalxr  45969  xlimpnfxnegmnf  46000  liminfpnfuz  46002  xlimpnfxnegmnf2  46044