Theorem caovdig 5632

Description: Convert an operation distributive law to class notation. (Contributed by set.mm contributors, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)

Hypothesis

Ref	Expression
caovdig.1	⊢ ((φ ∧ (x ∈ S ∧ y ∈ S ∧ z ∈ S)) → (xG(yFz)) = ((xGy)F(xGz)))

Assertion

Ref	Expression
caovdig	⊢ ((φ ∧ (A ∈ S ∧ B ∈ S ∧ C ∈ S)) → (AG(BFC)) = ((AGB)F(AGC)))

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 caovdig

Step	Hyp	Ref	Expression
1		caovdig.1	. . 3 ⊢ ((φ ∧ (x ∈ S ∧ y ∈ S ∧ z ∈ S)) → (xG(yFz)) = ((xGy)F(xGz)))
2	1	ralrimivvva 2707	. 2 ⊢ (φ → ∀x ∈ S ∀y ∈ S ∀z ∈ S (xG(yFz)) = ((xGy)F(xGz)))
3		oveq1 5530	. . . 4 ⊢ (x = A → (xG(yFz)) = (AG(yFz)))
4		oveq1 5530	. . . . 5 ⊢ (x = A → (xGy) = (AGy))
5		oveq1 5530	. . . . 5 ⊢ (x = A → (xGz) = (AGz))
6	4, 5	oveq12d 5540	. . . 4 ⊢ (x = A → ((xGy)F(xGz)) = ((AGy)F(AGz)))
7	3, 6	eqeq12d 2367	. . 3 ⊢ (x = A → ((xG(yFz)) = ((xGy)F(xGz)) ↔ (AG(yFz)) = ((AGy)F(AGz))))
8		oveq1 5530	. . . . 5 ⊢ (y = B → (yFz) = (BFz))
9	8	oveq2d 5538	. . . 4 ⊢ (y = B → (AG(yFz)) = (AG(BFz)))
10		oveq2 5531	. . . . 5 ⊢ (y = B → (AGy) = (AGB))
11	10	oveq1d 5537	. . . 4 ⊢ (y = B → ((AGy)F(AGz)) = ((AGB)F(AGz)))
12	9, 11	eqeq12d 2367	. . 3 ⊢ (y = B → ((AG(yFz)) = ((AGy)F(AGz)) ↔ (AG(BFz)) = ((AGB)F(AGz))))
13		oveq2 5531	. . . . 5 ⊢ (z = C → (BFz) = (BFC))
14	13	oveq2d 5538	. . . 4 ⊢ (z = C → (AG(BFz)) = (AG(BFC)))
15		oveq2 5531	. . . . 5 ⊢ (z = C → (AGz) = (AGC))
16	15	oveq2d 5538	. . . 4 ⊢ (z = C → ((AGB)F(AGz)) = ((AGB)F(AGC)))
17	14, 16	eqeq12d 2367	. . 3 ⊢ (z = C → ((AG(BFz)) = ((AGB)F(AGz)) ↔ (AG(BFC)) = ((AGB)F(AGC))))
18	7, 12, 17	rspc3v 2964	. 2 ⊢ ((A ∈ S ∧ B ∈ S ∧ C ∈ S) → (∀x ∈ S ∀y ∈ S ∀z ∈ S (xG(yFz)) = ((xGy)F(xGz)) → (AG(BFC)) = ((AGB)F(AGC))))
19	2, 18	mpan9 455	1 ⊢ ((φ ∧ (A ∈ S ∧ B ∈ S ∧ C ∈ S)) → (AG(BFC)) = ((AGB)F(AGC)))

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-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-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:  caovdi  5634