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Theorem caovdir 5642
 Description: Reverse distributive law. (Contributed by set.mm contributors, 26-Aug-1995.)
Hypotheses
Ref Expression
caoprd.1 A V
caoprd.2 B V
caoprd.3 C V
caoprd.com (xGy) = (yGx)
caoprd.distr (xG(yFz)) = ((xGy)F(xGz))
Assertion
Ref Expression
caovdir ((AFB)GC) = ((AGC)F(BGC))
Distinct variable groups:   x,y,z,F   x,A,y,z   x,B,y,z   x,C,y,z   x,G,y,z

Proof of Theorem caovdir
StepHypRef Expression
1 caoprd.3 . . 3 C V
2 caoprd.1 . . 3 A V
3 caoprd.2 . . 3 B V
4 caoprd.distr . . 3 (xG(yFz)) = ((xGy)F(xGz))
51, 2, 3, 4caovdi 5634 . 2 (CG(AFB)) = ((CGA)F(CGB))
6 ovex 5551 . . 3 (AFB) V
7 caoprd.com . . 3 (xGy) = (yGx)
81, 6, 7caovcom 5625 . 2 (CG(AFB)) = ((AFB)GC)
91, 2, 7caovcom 5625 . . 3 (CGA) = (AGC)
101, 3, 7caovcom 5625 . . 3 (CGB) = (BGC)
119, 10oveq12i 5535 . 2 ((CGA)F(CGB)) = ((AGC)F(BGC))
125, 8, 113eqtr3i 2381 1 ((AFB)GC) = ((AGC)F(BGC))
 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:  caovdilem  5643
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