Theorem nfxneg 46292

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 46272	. 2 ⊢ (⊤ → Ⅎ𝑥-𝑒𝐴)
4	3	mptru 1577	1 ⊢ Ⅎ𝑥-𝑒𝐴

This proof depends on syntax axioms:  ⊤wtru 1571  Ⅎwnfc 2907  -_𝑒cxne 13163

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-neg 11471  df-xneg 13166

This theorem is used by:  liminfvalxr  46614  xlimpnfxnegmnf  46645  liminfpnfuz  46647  xlimpnfxnegmnf2  46689