Theorem xrnegcon1d 11240

Description: Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)

Hypotheses

Ref	Expression
xrnegcon1d.a	|- ( ph -> A e. RR* )
xrnegcon1d.b	|- ( ph -> B e. RR* )

Assertion

Ref	Expression
xrnegcon1d	|- ( ph -> ( -e A = B <-> -e B = A ) )

Proof of Theorem xrnegcon1d

Step	Hyp	Ref	Expression
1		xrnegcon1d.b	. . . 4 |- ( ph -> B e. RR* )
2		xnegneg 9804	. . . . 5 |- ( B e. RR* -> -e -e B = B )
3	2	eqeq2d 2187	. . . 4 |- ( B e. RR* -> ( -e A = -e -e B <-> -e A = B ) )
4	1, 3	syl 14	. . 3 |- ( ph -> ( -e A = -e -e B <-> -e A = B ) )
5		xrnegcon1d.a	. . . 4 |- ( ph -> A e. RR* )
6	1	xnegcld 9826	. . . 4 |- ( ph -> -e B e. RR* )
7		xneg11 9805	. . . 4 |- ( ( A e. RR* /\ -e B e. RR* ) -> ( -e A = -e -e B <-> A = -e B ) )
8	5, 6, 7	syl2anc 411	. . 3 |- ( ph -> ( -e A = -e -e B <-> A = -e B ) )
9	4, 8	bitr3d 190	. 2 |- ( ph -> ( -e A = B <-> A = -e B ) )
10		eqcom 2177	. 2 |- ( A = -e B <-> -e B = A )
11	9, 10	bitrdi 196	1 |- ( ph -> ( -e A = B <-> -e B = A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2146   RR*cxr 7965   -ecxne 9740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-sub 8104  df-neg 8105  df-xneg 9743

This theorem is referenced by:  xrminmax  11241  xrmineqinf  11245