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Theorem caovdirg 5633
 Description: Convert an operation reverse distributive law to class notation. (Contributed by set.mm contributors, 19-Oct-2014.)
Hypothesis
Ref Expression
caovdirg.1 ((φ (x S y S z S)) → ((xFy)Gz) = ((xGz)F(yGz)))
Assertion
Ref Expression
caovdirg ((φ (A S B S C S)) → ((AFB)GC) = ((AGC)F(BGC)))
Distinct variable groups:   x,y,z,F   x,S,y,z   x,A,y,z   x,B,y,z   x,C,y,z   x,G,y,z   φ,x,y,z

Proof of Theorem caovdirg
StepHypRef Expression
1 caovdirg.1 . . 3 ((φ (x S y S z S)) → ((xFy)Gz) = ((xGz)F(yGz)))
21ralrimivvva 2707 . 2 (φx S y S z S ((xFy)Gz) = ((xGz)F(yGz)))
3 oveq1 5530 . . . . 5 (x = A → (xFy) = (AFy))
43oveq1d 5537 . . . 4 (x = A → ((xFy)Gz) = ((AFy)Gz))
5 oveq1 5530 . . . . 5 (x = A → (xGz) = (AGz))
65oveq1d 5537 . . . 4 (x = A → ((xGz)F(yGz)) = ((AGz)F(yGz)))
74, 6eqeq12d 2367 . . 3 (x = A → (((xFy)Gz) = ((xGz)F(yGz)) ↔ ((AFy)Gz) = ((AGz)F(yGz))))
8 oveq2 5531 . . . . 5 (y = B → (AFy) = (AFB))
98oveq1d 5537 . . . 4 (y = B → ((AFy)Gz) = ((AFB)Gz))
10 oveq1 5530 . . . . 5 (y = B → (yGz) = (BGz))
1110oveq2d 5538 . . . 4 (y = B → ((AGz)F(yGz)) = ((AGz)F(BGz)))
129, 11eqeq12d 2367 . . 3 (y = B → (((AFy)Gz) = ((AGz)F(yGz)) ↔ ((AFB)Gz) = ((AGz)F(BGz))))
13 oveq2 5531 . . . 4 (z = C → ((AFB)Gz) = ((AFB)GC))
14 oveq2 5531 . . . . 5 (z = C → (AGz) = (AGC))
15 oveq2 5531 . . . . 5 (z = C → (BGz) = (BGC))
1614, 15oveq12d 5540 . . . 4 (z = C → ((AGz)F(BGz)) = ((AGC)F(BGC)))
1713, 16eqeq12d 2367 . . 3 (z = C → (((AFB)Gz) = ((AGz)F(BGz)) ↔ ((AFB)GC) = ((AGC)F(BGC))))
187, 12, 17rspc3v 2964 . 2 ((A S B S C S) → (x S y S z S ((xFy)Gz) = ((xGz)F(yGz)) → ((AFB)GC) = ((AGC)F(BGC))))
192, 18mpan9 455 1 ((φ (A S B S C S)) → ((AFB)GC) = ((AGC)F(BGC)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614  (class class class)co 5525 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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: (None)
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