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Theorem caovcld 5596
 Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovclg.1 ((φ (x 𝐶 y 𝐷)) → (x𝐹y) 𝐸)
caovcld.2 (φA 𝐶)
caovcld.3 (φB 𝐷)
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 (φA 𝐶)
3 caovcld.3 . 2 (φB 𝐷)
4 caovclg.1 . . 3 ((φ (x 𝐶 y 𝐷)) → (x𝐹y) 𝐸)
54caovclg 5595 . 2 ((φ (A 𝐶 B 𝐷)) → (A𝐹B) 𝐸)
61, 2, 3, 5syl12anc 1132 1 (φ → (A𝐹B) 𝐸)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ 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:  caovdir2d  5619  caov4d  5627  caovdilemd  5634  caovlem2d  5635  grprinvd  5638  ecopovtrn  6139  ecopovtrng  6142  ordpipqqs  6358  ltanqg  6384  ltmnqg  6385  recexprlem1ssu  6605  mulgt0sr  6684  mulextsr1lem  6686  axmulass  6737  frec2uzrdg  8856  frecuzrdgsuc  8862  iseqovex  8879  iseqval  8880  iseqp1  8884
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