xneg11 - Intuitionistic Logic Explorer

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Theorem xneg11 8747

Description: Extended real version of neg11 7262. (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 8740	. . 3 ⊢ (-_𝑒𝐴 = -_𝑒𝐵 → -_𝑒-_𝑒𝐴 = -_𝑒-_𝑒𝐵)
2		xnegneg 8746	. . . 4 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
3		xnegneg 8746	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
4	2, 3	eqeqan12d 2055	. . 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 8740	. 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 1243 ∈ wcel 1393 ℝ^*cxr 7059 -_𝑒cxne 8686

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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995

This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 df-sub 7184 df-neg 7185 df-xneg 8689

This theorem is referenced by: (None)