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Theorem addcan2ad 7198
Description: Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 7196. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
addcand.1 (𝜑𝐴 ∈ ℂ)
addcand.2 (𝜑𝐵 ∈ ℂ)
addcand.3 (𝜑𝐶 ∈ ℂ)
addcan2ad.4 (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶))
Assertion
Ref Expression
addcan2ad (𝜑𝐴 = 𝐵)

Proof of Theorem addcan2ad
StepHypRef Expression
1 addcan2ad.4 . 2 (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶))
2 addcand.1 . . 3 (𝜑𝐴 ∈ ℂ)
3 addcand.2 . . 3 (𝜑𝐵 ∈ ℂ)
4 addcand.3 . . 3 (𝜑𝐶 ∈ ℂ)
52, 3, 4addcan2d 7196 . 2 (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
61, 5mpbid 135 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  (class class class)co 5512  cc 6887   + caddc 6892
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-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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  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-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515
This theorem is referenced by: (None)
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