Theorem addc6 4419

Description: Rearrange cardinal summation of six arguments. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
addc6	|- (((A +o B) +o (C +o D)) +o (E +o F)) = (((A +o C) +o E) +o ((B +o D) +o F))

Proof of Theorem addc6

Step	Hyp	Ref	Expression
1		addc4 4418	. . . . 5 |- ((A +o B) +o (C +o D)) = ((A +o C) +o (B +o D))
2	1	addceq1i 4387	. . . 4 |- (((A +o B) +o (C +o D)) +o E) = (((A +o C) +o (B +o D)) +o E)
3		addc32 4417	. . . 4 |- (((A +o C) +o (B +o D)) +o E) = (((A +o C) +o E) +o (B +o D))
4	2, 3	eqtri 2373	. . 3 |- (((A +o B) +o (C +o D)) +o E) = (((A +o C) +o E) +o (B +o D))
5	4	addceq1i 4387	. 2 |- ((((A +o B) +o (C +o D)) +o E) +o F) = ((((A +o C) +o E) +o (B +o D)) +o F)
6		addcass 4416	. 2 |- ((((A +o B) +o (C +o D)) +o E) +o F) = (((A +o B) +o (C +o D)) +o (E +o F))
7		addcass 4416	. 2 |- ((((A +o C) +o E) +o (B +o D)) +o F) = (((A +o C) +o E) +o ((B +o D) +o F))
8	5, 6, 7	3eqtr3i 2381	1 |- (((A +o B) +o (C +o D)) +o (E +o F)) = (((A +o C) +o E) +o ((B +o D) +o F))

Syntax hints:   = wceq 1642   +o cplc 4376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-sik 4193  df-ssetk 4194  df-addc 4379

This theorem is referenced by:  nnc3n3p1  6279