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Theorem int-addassocd 39254
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 10358 . . 3 (𝜑𝐴 ∈ ℂ)
3 int-addassocd.2 . . . 4 (𝜑𝐶 ∈ ℝ)
43recnd 10358 . . 3 (𝜑𝐶 ∈ ℂ)
5 int-addassocd.3 . . . 4 (𝜑𝐷 ∈ ℝ)
65recnd 10358 . . 3 (𝜑𝐷 ∈ ℂ)
72, 4, 6addassd 10352 . 2 (𝜑 → ((𝐴 + 𝐶) + 𝐷) = (𝐴 + (𝐶 + 𝐷)))
8 int-addassocd.4 . . 3 (𝜑𝐴 = 𝐵)
98oveq1d 6894 . 2 (𝜑 → (𝐴 + (𝐶 + 𝐷)) = (𝐵 + (𝐶 + 𝐷)))
107, 9eqtr2d 2835 1 (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  (class class class)co 6879  cr 10224   + caddc 10228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-resscn 10282  ax-addass 10290
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-iota 6065  df-fv 6110  df-ov 6882
This theorem is referenced by: (None)
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