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Theorem int-addassocd 42539
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 11191 . . 3 (𝜑𝐴 ∈ ℂ)
3 int-addassocd.2 . . . 4 (𝜑𝐶 ∈ ℝ)
43recnd 11191 . . 3 (𝜑𝐶 ∈ ℂ)
5 int-addassocd.3 . . . 4 (𝜑𝐷 ∈ ℝ)
65recnd 11191 . . 3 (𝜑𝐷 ∈ ℂ)
72, 4, 6addassd 11185 . 2 (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷)))
8 int-addassocd.4 . . 3 (𝜑𝐴 = 𝐵)
98oveq1d 7376 . 2 (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷)))
107, 9eqtr2d 2774 1 (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  (class class class)co 7361  cr 11058   + caddc 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-resscn 11116  ax-addass 11124
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364
This theorem is referenced by: (None)
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