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Mirrors > Home > ILE Home > Th. List > negcon1 | GIF version |
Description: Negative contraposition law. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
negcon1 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 8171 | . . . 4 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | neg11 8222 | . . . 4 ⊢ ((-𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (--𝐴 = -𝐵 ↔ -𝐴 = 𝐵)) | |
3 | 1, 2 | sylan 283 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (--𝐴 = -𝐵 ↔ -𝐴 = 𝐵)) |
4 | negneg 8221 | . . . . 5 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
5 | 4 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → --𝐴 = 𝐴) |
6 | 5 | eqeq1d 2196 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (--𝐴 = -𝐵 ↔ 𝐴 = -𝐵)) |
7 | 3, 6 | bitr3d 190 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ 𝐴 = -𝐵)) |
8 | eqcom 2189 | . 2 ⊢ (𝐴 = -𝐵 ↔ -𝐵 = 𝐴) | |
9 | 7, 8 | bitrdi 196 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 ℂcc 7823 -cneg 8143 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-resscn 7917 ax-1cn 7918 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8144 df-neg 8145 |
This theorem is referenced by: negcon2 8224 negcon1i 8253 negcon1d 8276 elznn0 9282 qsqeqor 10645 |
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