xneg11 - Intuitionistic Logic Explorer

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Theorem xneg11 8901

Description: Extended real version of neg11 7359. (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 8894	. . 3 ⊢ (-_𝑒𝐴 = -_𝑒𝐵 → -_𝑒-_𝑒𝐴 = -_𝑒-_𝑒𝐵)
2		xnegneg 8900	. . . 4 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
3		xnegneg 8900	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
4	2, 3	eqeqan12d 2096	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒-_𝑒𝐴 = -_𝑒-_𝑒𝐵 ↔ 𝐴 = 𝐵))
5	1, 4	syl5ib 152	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒𝐵 → 𝐴 = 𝐵))
6		xnegeq 8894	. 2 ⊢ (𝐴 = 𝐵 → -_𝑒𝐴 = -_𝑒𝐵)
7	5, 6	impbid1 140	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒𝐵 ↔ 𝐴 = 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ℝ^*cxr 7152 -_𝑒cxne 8840

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087

This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-sub 7281 df-neg 7282 df-xneg 8843

This theorem is referenced by: (None)