Theorem nfxneg 45464

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 45444	. 2 ⊢ (⊤ → Ⅎ𝑥-𝑒𝐴)
4	3	mptru 1547	1 ⊢ Ⅎ𝑥-𝑒𝐴

Syntax hints:  ⊤wtru 1541  Ⅎwnfc 2877  -_𝑒cxne 13076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-neg 11415  df-xneg 13079

This theorem is referenced by:  liminfvalxr  45788  xlimpnfxnegmnf  45819  liminfpnfuz  45821  xlimpnfxnegmnf2  45863