Theorem nfxneg 45569

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 45549	. 2 ⊢ (⊤ → Ⅎ𝑥-𝑒𝐴)
4	3	mptru 1548	1 ⊢ Ⅎ𝑥-𝑒𝐴

Syntax hints:  ⊤wtru 1542  Ⅎwnfc 2879  -_𝑒cxne 13008

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-neg 11347  df-xneg 13011

This theorem is referenced by:  liminfvalxr  45891  xlimpnfxnegmnf  45922  liminfpnfuz  45924  xlimpnfxnegmnf2  45966