ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovcld GIF version

Theorem caovcld 6030
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovclg.1 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)
caovcld.2 (𝜑𝐴𝐶)
caovcld.3 (𝜑𝐵𝐷)
Assertion
Ref Expression
caovcld (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐸,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem caovcld
StepHypRef Expression
1 id 19 . 2 (𝜑𝜑)
2 caovcld.2 . 2 (𝜑𝐴𝐶)
3 caovcld.3 . 2 (𝜑𝐵𝐷)
4 caovclg.1 . . 3 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)
54caovclg 6029 . 2 ((𝜑 ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
61, 2, 3, 5syl12anc 1236 1 (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148  (class class class)co 5877
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 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880
This theorem is referenced by:  caovdir2d  6053  caov4d  6061  caovdilemd  6068  caovlem2d  6069  ecopovtrn  6634  ecopovtrng  6637  ordpipqqs  7375  ltanqg  7401  ltmnqg  7402  recexprlem1ssu  7635  mulgt0sr  7779  mulextsr1lem  7781  axmulass  7874  frec2uzrdg  10411  frecuzrdgsuc  10416  frecuzrdgsuctlem  10425  iseqovex  10458  seq3val  10460  seqf  10463  seq3p1  10464  seqp1cd  10468  seq3clss  10469  seq3distr  10515  climcn2  11319  qusaddvallemg  12757  grprinvd  12810
  Copyright terms: Public domain W3C validator