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Theorem add4 6949
Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
add4 (((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (B + 𝐷)))

Proof of Theorem add4
StepHypRef Expression
1 add12 6946 . . . . 5 ((B 𝐶 𝐷 ℂ) → (B + (𝐶 + 𝐷)) = (𝐶 + (B + 𝐷)))
213expb 1104 . . . 4 ((B (𝐶 𝐷 ℂ)) → (B + (𝐶 + 𝐷)) = (𝐶 + (B + 𝐷)))
32oveq2d 5471 . . 3 ((B (𝐶 𝐷 ℂ)) → (A + (B + (𝐶 + 𝐷))) = (A + (𝐶 + (B + 𝐷))))
43adantll 445 . 2 (((A B ℂ) (𝐶 𝐷 ℂ)) → (A + (B + (𝐶 + 𝐷))) = (A + (𝐶 + (B + 𝐷))))
5 addcl 6784 . . 3 ((𝐶 𝐷 ℂ) → (𝐶 + 𝐷) ℂ)
6 addass 6789 . . . 4 ((A B (𝐶 + 𝐷) ℂ) → ((A + B) + (𝐶 + 𝐷)) = (A + (B + (𝐶 + 𝐷))))
763expa 1103 . . 3 (((A B ℂ) (𝐶 + 𝐷) ℂ) → ((A + B) + (𝐶 + 𝐷)) = (A + (B + (𝐶 + 𝐷))))
85, 7sylan2 270 . 2 (((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = (A + (B + (𝐶 + 𝐷))))
9 addcl 6784 . . . 4 ((B 𝐷 ℂ) → (B + 𝐷) ℂ)
10 addass 6789 . . . . 5 ((A 𝐶 (B + 𝐷) ℂ) → ((A + 𝐶) + (B + 𝐷)) = (A + (𝐶 + (B + 𝐷))))
11103expa 1103 . . . 4 (((A 𝐶 ℂ) (B + 𝐷) ℂ) → ((A + 𝐶) + (B + 𝐷)) = (A + (𝐶 + (B + 𝐷))))
129, 11sylan2 270 . . 3 (((A 𝐶 ℂ) (B 𝐷 ℂ)) → ((A + 𝐶) + (B + 𝐷)) = (A + (𝐶 + (B + 𝐷))))
1312an4s 522 . 2 (((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + 𝐶) + (B + 𝐷)) = (A + (𝐶 + (B + 𝐷))))
144, 8, 133eqtr4d 2079 1 (((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (B + 𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  (class class class)co 5455  cc 6689   + caddc 6694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-addcl 6759  ax-addcom 6763  ax-addass 6765
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  add42  6950  add4i  6953  add4d  6957
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