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Theorem caov4 5639
 Description: Rearrange arguments in a commutative, associative operation. (Contributed by set.mm contributors, 26-Aug-1995.)
Hypotheses
Ref Expression
caopr.1 A V
caopr.2 B V
caopr.3 C V
caopr.com (xFy) = (yFx)
caopr.ass ((xFy)Fz) = (xF(yFz))
caopr.4 D V
Assertion
Ref Expression
caov4 ((AFB)F(CFD)) = ((AFC)F(BFD))
Distinct variable groups:   x,y,z,F   x,A,y,z   x,B,y,z   x,C,y,z   x,D,y,z

Proof of Theorem caov4
StepHypRef Expression
1 caopr.2 . . . 4 B V
2 caopr.3 . . . 4 C V
3 caopr.4 . . . 4 D V
4 caopr.com . . . 4 (xFy) = (yFx)
5 caopr.ass . . . 4 ((xFy)Fz) = (xF(yFz))
61, 2, 3, 4, 5caov12 5636 . . 3 (BF(CFD)) = (CF(BFD))
76oveq2i 5534 . 2 (AF(BF(CFD))) = (AF(CF(BFD)))
8 caopr.1 . . 3 A V
9 ovex 5551 . . 3 (CFD) V
108, 1, 9, 5caovass 5627 . 2 ((AFB)F(CFD)) = (AF(BF(CFD)))
11 ovex 5551 . . 3 (BFD) V
128, 2, 11, 5caovass 5627 . 2 ((AFC)F(BFD)) = (AF(CF(BFD)))
137, 10, 123eqtr4i 2383 1 ((AFB)F(CFD)) = ((AFC)F(BFD))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2859  (class class class)co 5525 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795  df-ov 5526 This theorem is referenced by:  caov42  5641
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