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Theorem addcan2ad 7439
Description: Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 7437. (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 7437 . 2 (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
61, 5mpbid 145 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  (class class class)co 5565  cc 7118   + caddc 7123
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-resscn 7207  ax-1cn 7208  ax-icn 7210  ax-addcl 7211  ax-addrcl 7212  ax-mulcl 7213  ax-addcom 7215  ax-addass 7217  ax-distr 7219  ax-i2m1 7220  ax-0id 7223  ax-rnegex 7224  ax-cnre 7226
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-iota 4918  df-fv 4961  df-ov 5568
This theorem is referenced by: (None)
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