Theorem nfxneg 43001

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 42981	. 2 ⊢ (⊤ → Ⅎ𝑥-𝑒𝐴)
4	3	mptru 1546	1 ⊢ Ⅎ𝑥-𝑒𝐴

Syntax hints:  ⊤wtru 1540  Ⅎwnfc 2887  -_𝑒cxne 12845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-neg 11208  df-xneg 12848

This theorem is referenced by:  liminfvalxr  43324  xlimpnfxnegmnf  43355  liminfpnfuz  43357  xlimpnfxnegmnf2  43399