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Mirrors > Home > ILE Home > Th. List > addcan2ad | GIF version |
Description: Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 7293. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
addcand.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addcand.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
addcan2ad.4 | ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶)) |
Ref | Expression |
---|---|
addcan2ad | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcan2ad.4 | . 2 ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶)) | |
2 | addcand.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | addcand.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | addcand.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
5 | 2, 3, 4 | addcan2d 7293 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) |
6 | 1, 5 | mpbid 145 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 (class class class)co 5532 ℂcc 6979 + caddc 6984 |
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-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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 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-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: (None) |
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