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Theorem caovcan 5628
 Description: Convert an operation cancellation law to class notation. (Contributed by set.mm contributors, 20-Aug-1995.)
Hypotheses
Ref Expression
caovcan.1 C V
caovcan.2 ((x S y S) → ((xFy) = (xFz) → y = z))
Assertion
Ref Expression
caovcan ((A S B S) → ((AFB) = (AFC) → B = C))
Distinct variable groups:   x,y,z,F   x,S,y,z   x,A,y,z   x,B,y,z   x,C,y,z

Proof of Theorem caovcan
StepHypRef Expression
1 oveq1 5530 . . . 4 (x = A → (xFy) = (AFy))
2 oveq1 5530 . . . 4 (x = A → (xFC) = (AFC))
31, 2eqeq12d 2367 . . 3 (x = A → ((xFy) = (xFC) ↔ (AFy) = (AFC)))
43imbi1d 308 . 2 (x = A → (((xFy) = (xFC) → y = C) ↔ ((AFy) = (AFC) → y = C)))
5 oveq2 5531 . . . 4 (y = B → (AFy) = (AFB))
65eqeq1d 2361 . . 3 (y = B → ((AFy) = (AFC) ↔ (AFB) = (AFC)))
7 eqeq1 2359 . . 3 (y = B → (y = CB = C))
86, 7imbi12d 311 . 2 (y = B → (((AFy) = (AFC) → y = C) ↔ ((AFB) = (AFC) → B = C)))
9 caovcan.1 . . 3 C V
10 oveq2 5531 . . . . . 6 (z = C → (xFz) = (xFC))
1110eqeq2d 2364 . . . . 5 (z = C → ((xFy) = (xFz) ↔ (xFy) = (xFC)))
12 eqeq2 2362 . . . . 5 (z = C → (y = zy = C))
1311, 12imbi12d 311 . . . 4 (z = C → (((xFy) = (xFz) → y = z) ↔ ((xFy) = (xFC) → y = C)))
1413imbi2d 307 . . 3 (z = C → (((x S y S) → ((xFy) = (xFz) → y = z)) ↔ ((x S y S) → ((xFy) = (xFC) → y = C))))
15 caovcan.2 . . 3 ((x S y S) → ((xFy) = (xFz) → y = z))
169, 14, 15vtocl 2909 . 2 ((x S y S) → ((xFy) = (xFC) → y = C))
174, 8, 16vtocl2ga 2922 1 ((A S B S) → ((AFB) = (AFC) → B = C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = 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-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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