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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 7863. (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 7863 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) |
6 | 1, 5 | mpbid 146 | 1 ⊢ (𝜑 → 𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 ∈ wcel 1461 (class class class)co 5726 ℂcc 7539 + caddc 7544 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-resscn 7631 ax-1cn 7632 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-iota 5044 df-fv 5087 df-ov 5729 |
This theorem is referenced by: divalglemqt 11458 |
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