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Theorem caovcanrd 5606
Description: Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovcang.1 ((φ (x 𝑇 y 𝑆 z 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z))
caovcand.2 (φA 𝑇)
caovcand.3 (φB 𝑆)
caovcand.4 (φ𝐶 𝑆)
caovcanrd.5 (φA 𝑆)
caovcanrd.6 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
Assertion
Ref Expression
caovcanrd (φ → ((B𝐹A) = (𝐶𝐹A) ↔ 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 caovcanrd
StepHypRef Expression
1 caovcanrd.6 . . . 4 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
2 caovcanrd.5 . . . 4 (φA 𝑆)
3 caovcand.3 . . . 4 (φB 𝑆)
41, 2, 3caovcomd 5599 . . 3 (φ → (A𝐹B) = (B𝐹A))
5 caovcand.4 . . . 4 (φ𝐶 𝑆)
61, 2, 5caovcomd 5599 . . 3 (φ → (A𝐹𝐶) = (𝐶𝐹A))
74, 6eqeq12d 2051 . 2 (φ → ((A𝐹B) = (A𝐹𝐶) ↔ (B𝐹A) = (𝐶𝐹A)))
8 caovcang.1 . . 3 ((φ (x 𝑇 y 𝑆 z 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z))
9 caovcand.2 . . 3 (φA 𝑇)
108, 9, 3, 5caovcand 5605 . 2 (φ → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶))
117, 10bitr3d 179 1 (φ → ((B𝐹A) = (𝐶𝐹A) ↔ B = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  (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: (None)
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