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Theorem caovcang 5604
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypothesis
Ref Expression
caovcang.1 ((φ (x 𝑇 y 𝑆 z 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z))
Assertion
Ref Expression
caovcang ((φ (A 𝑇 B 𝑆 𝐶 𝑆)) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   φ,x,y,z   x,𝐹,y,z   x,𝑆,y,z   x,𝑇,y,z

Proof of Theorem caovcang
StepHypRef Expression
1 caovcang.1 . . 3 ((φ (x 𝑇 y 𝑆 z 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z))
21ralrimivvva 2396 . 2 (φx 𝑇 y 𝑆 z 𝑆 ((x𝐹y) = (x𝐹z) ↔ y = z))
3 oveq1 5462 . . . . 5 (x = A → (x𝐹y) = (A𝐹y))
4 oveq1 5462 . . . . 5 (x = A → (x𝐹z) = (A𝐹z))
53, 4eqeq12d 2051 . . . 4 (x = A → ((x𝐹y) = (x𝐹z) ↔ (A𝐹y) = (A𝐹z)))
65bibi1d 222 . . 3 (x = A → (((x𝐹y) = (x𝐹z) ↔ y = z) ↔ ((A𝐹y) = (A𝐹z) ↔ y = z)))
7 oveq2 5463 . . . . 5 (y = B → (A𝐹y) = (A𝐹B))
87eqeq1d 2045 . . . 4 (y = B → ((A𝐹y) = (A𝐹z) ↔ (A𝐹B) = (A𝐹z)))
9 eqeq1 2043 . . . 4 (y = B → (y = zB = z))
108, 9bibi12d 224 . . 3 (y = B → (((A𝐹y) = (A𝐹z) ↔ y = z) ↔ ((A𝐹B) = (A𝐹z) ↔ B = z)))
11 oveq2 5463 . . . . 5 (z = 𝐶 → (A𝐹z) = (A𝐹𝐶))
1211eqeq2d 2048 . . . 4 (z = 𝐶 → ((A𝐹B) = (A𝐹z) ↔ (A𝐹B) = (A𝐹𝐶)))
13 eqeq2 2046 . . . 4 (z = 𝐶 → (B = zB = 𝐶))
1412, 13bibi12d 224 . . 3 (z = 𝐶 → (((A𝐹B) = (A𝐹z) ↔ B = z) ↔ ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶)))
156, 10, 14rspc3v 2659 . 2 ((A 𝑇 B 𝑆 𝐶 𝑆) → (x 𝑇 y 𝑆 z 𝑆 ((x𝐹y) = (x𝐹z) ↔ y = z) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶)))
162, 15mpan9 265 1 ((φ (A 𝑇 B 𝑆 𝐶 𝑆)) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wral 2300  (class class class)co 5455
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  caovcand  5605
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