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Theorem caovcan 5584
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 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,A   x,B,y,z   x,𝐶,y,z   x,𝐹,y,z   x,𝑆,y,z

Proof of Theorem caovcan
StepHypRef Expression
1 oveq1 5439 . . . 4 (x = A → (x𝐹y) = (A𝐹y))
2 oveq1 5439 . . . 4 (x = A → (x𝐹𝐶) = (A𝐹𝐶))
31, 2eqeq12d 2032 . . 3 (x = A → ((x𝐹y) = (x𝐹𝐶) ↔ (A𝐹y) = (A𝐹𝐶)))
43imbi1d 220 . 2 (x = A → (((x𝐹y) = (x𝐹𝐶) → y = 𝐶) ↔ ((A𝐹y) = (A𝐹𝐶) → y = 𝐶)))
5 oveq2 5440 . . . 4 (y = B → (A𝐹y) = (A𝐹B))
65eqeq1d 2026 . . 3 (y = B → ((A𝐹y) = (A𝐹𝐶) ↔ (A𝐹B) = (A𝐹𝐶)))
7 eqeq1 2024 . . 3 (y = B → (y = 𝐶B = 𝐶))
86, 7imbi12d 223 . 2 (y = B → (((A𝐹y) = (A𝐹𝐶) → y = 𝐶) ↔ ((A𝐹B) = (A𝐹𝐶) → B = 𝐶)))
9 caovcan.1 . . 3 𝐶 V
10 oveq2 5440 . . . . . 6 (z = 𝐶 → (x𝐹z) = (x𝐹𝐶))
1110eqeq2d 2029 . . . . 5 (z = 𝐶 → ((x𝐹y) = (x𝐹z) ↔ (x𝐹y) = (x𝐹𝐶)))
12 eqeq2 2027 . . . . 5 (z = 𝐶 → (y = zy = 𝐶))
1311, 12imbi12d 223 . . . 4 (z = 𝐶 → (((x𝐹y) = (x𝐹z) → y = z) ↔ ((x𝐹y) = (x𝐹𝐶) → y = 𝐶)))
1413imbi2d 219 . . 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 2581 . 2 ((x 𝑆 y 𝑆) → ((x𝐹y) = (x𝐹𝐶) → y = 𝐶))
174, 8, 16vtocl2ga 2594 1 ((A 𝑆 B 𝑆) → ((A𝐹B) = (A𝐹𝐶) → B = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226   wcel 1370  Vcvv 2531  (class class class)co 5432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-un 2895  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-iota 4790  df-fv 4833  df-ov 5435
This theorem is referenced by: (None)
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