Theorem add12d 8351

Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
addd.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
addd.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
addd.3	⊢ (𝜑 → 𝐶 ∈ ℂ)

Assertion

Ref	Expression
add12d	⊢ (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))

Proof of Theorem add12d

Step	Hyp	Ref	Expression
1		addd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		addd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		addd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		add12 8342	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))
5	1, 2, 3, 4	syl3anc 1273	1 ⊢ (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2201  (class class class)co 6023  ℂcc 8035   + caddc 8040

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212  ax-addcom 8137  ax-addass 8139

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-v 2803  df-un 3203  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-iota 5288  df-fv 5336  df-ov 6026

This theorem is referenced by:  subsub2  8412  bdtrilem  11822  pcadd2  12937