xneg11 - Intuitionistic Logic Explorer

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Theorem xneg11 8517

Description: Extended real version of neg11 7058. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xneg11	⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^*) → (-_𝑒A = -_𝑒B ↔ A = B))

**Proof of Theorem xneg11**
Step	Hyp	Ref	Expression
1		xnegeq 8510	. . 3 ⊢ (-_𝑒A = -_𝑒B → -_𝑒-_𝑒A = -_𝑒-_𝑒B)
2		xnegneg 8516	. . . 4 ⊢ (A ∈ ℝ^* → -_𝑒-_𝑒A = A)
3		xnegneg 8516	. . . 4 ⊢ (B ∈ ℝ^* → -_𝑒-_𝑒B = B)
4	2, 3	eqeqan12d 2052	. . 3 ⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^*) → (-_𝑒-_𝑒A = -_𝑒-_𝑒B ↔ A = B))
5	1, 4	syl5ib 143	. 2 ⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^*) → (-_𝑒A = -_𝑒B → A = B))
6		xnegeq 8510	. 2 ⊢ (A = B → -_𝑒A = -_𝑒B)
7	5, 6	impbid1 130	1 ⊢ ((A ∈ ℝ^* ∧ B ∈ ℝ^*) → (-_𝑒A = -_𝑒B ↔ A = B))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ℝ^*cxr 6856 -_𝑒cxne 8456

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-addass 6785 ax-distr 6787 ax-i2m1 6788 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794

This theorem depends on definitions: df-bi 110 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-if 3326 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-pnf 6859 df-mnf 6860 df-xr 6861 df-sub 6981 df-neg 6982 df-xneg 8459

This theorem is referenced by: (None)