Theorem addcan2ad 8456

Description: Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8454. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
addcand.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
addcand.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
addcand.3	⊢ (𝜑 → 𝐶 ∈ ℂ)
addcan2ad.4	⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶))

Assertion

Ref	Expression
addcan2ad	⊢ (𝜑 → 𝐴 = 𝐵)

Proof of Theorem addcan2ad

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 8454	. 2 ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
6	1, 5	mpbid 147	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  (class class class)co 6049  ℂcc 8121   + caddc 8126

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-resscn 8215  ax-1cn 8216  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-distr 8227  ax-i2m1 8228  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-ov 6052

This theorem is referenced by: (None)