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Theorem int-addassocd 44138
Description: AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
Hypotheses
Ref Expression
int-addassocd.1 (𝜑𝐴 ∈ ℝ)
int-addassocd.2 (𝜑𝐶 ∈ ℝ)
int-addassocd.3 (𝜑𝐷 ∈ ℝ)
int-addassocd.4 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
int-addassocd (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))

Proof of Theorem int-addassocd
StepHypRef Expression
1 int-addassocd.1 . . . 4 (𝜑𝐴 ∈ ℝ)
21recnd 11320 . . 3 (𝜑𝐴 ∈ ℂ)
3 int-addassocd.2 . . . 4 (𝜑𝐶 ∈ ℝ)
43recnd 11320 . . 3 (𝜑𝐶 ∈ ℂ)
5 int-addassocd.3 . . . 4 (𝜑𝐷 ∈ ℝ)
65recnd 11320 . . 3 (𝜑𝐷 ∈ ℂ)
72, 4, 6addassd 11314 . 2 (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷)))
8 int-addassocd.4 . . 3 (𝜑𝐴 = 𝐵)
98oveq1d 7465 . 2 (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷)))
107, 9eqtr2d 2781 1 (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  (class class class)co 7450  cr 11185   + caddc 11189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-resscn 11243  ax-addass 11251
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-ov 7453
This theorem is referenced by: (None)
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