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Mirrors > Home > ILE Home > Th. List > addcanad | GIF version |
Description: Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 8144. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
addcand.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addcand.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
addcanad.4 | ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐴 + 𝐶)) |
Ref | Expression |
---|---|
addcanad | ⊢ (𝜑 → 𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcanad.4 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐴 + 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 | addcand 8144 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) |
6 | 1, 5 | mpbid 147 | 1 ⊢ (𝜑 → 𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 (class class class)co 5878 ℂcc 7812 + caddc 7817 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-resscn 7906 ax-1cn 7907 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-iota 5180 df-fv 5226 df-ov 5881 |
This theorem is referenced by: divalglemqt 11927 |
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