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Theorem caovcanrd 6037
Description: Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovcang.1 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
caovcand.2 (𝜑𝐴𝑇)
caovcand.3 (𝜑𝐵𝑆)
caovcand.4 (𝜑𝐶𝑆)
caovcanrd.5 (𝜑𝐴𝑆)
caovcanrd.6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (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 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
2 caovcanrd.5 . . . 4 (𝜑𝐴𝑆)
3 caovcand.3 . . . 4 (𝜑𝐵𝑆)
41, 2, 3caovcomd 6030 . . 3 (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
5 caovcand.4 . . . 4 (𝜑𝐶𝑆)
61, 2, 5caovcomd 6030 . . 3 (𝜑 → (𝐴𝐹𝐶) = (𝐶𝐹𝐴))
74, 6eqeq12d 2192 . 2 (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ (𝐵𝐹𝐴) = (𝐶𝐹𝐴)))
8 caovcang.1 . . 3 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
9 caovcand.2 . . 3 (𝜑𝐴𝑇)
108, 9, 3, 5caovcand 6036 . 2 (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
117, 10bitr3d 190 1 (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  (class class class)co 5874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224  df-ov 5877
This theorem is referenced by: (None)
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