Theorem nfxnegd 39668

Description: Deduction version of nfxneg 39691. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
nfxnegd.1	|- ( ph -> F/_ x A )

Assertion

Ref	Expression
nfxnegd	|- ( ph -> F/_ x -e A )

Proof of Theorem nfxnegd

Step	Hyp	Ref	Expression
1		df-xneg 11946	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
2		nfxnegd.1	. . . 4 |- ( ph -> F/_ x A )
3		nfcvd 2765	. . . 4 |- ( ph -> F/_ x +oo )
4	2, 3	nfeqd 2772	. . 3 |- ( ph -> F/ x A = +oo )
5		nfcvd 2765	. . 3 |- ( ph -> F/_ x -oo )
6	2, 5	nfeqd 2772	. . . 4 |- ( ph -> F/ x A = -oo )
7	2	nfnegd 10276	. . . 4 |- ( ph -> F/_ x -u A )
8	6, 3, 7	nfifd 4114	. . 3 |- ( ph -> F/_ x if ( A = -oo , +oo , -u A ) )
9	4, 5, 8	nfifd 4114	. 2 |- ( ph -> F/_ x if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) ) )
10	1, 9	nfcxfrd 2763	1 |- ( ph -> F/_ x -e A )

Syntax hints:   -> wi 4   = wceq 1483   F/_wnfc 2751   ifcif 4086   +oocpnf 10071   -oocmnf 10072   -ucneg 10267   -ecxne 11943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-neg 10269  df-xneg 11946

This theorem is referenced by:  nfxneg  39691