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Theorem caovdird 5618
Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovdirg.1 ((φ (x 𝑆 y 𝑆 z 𝐾)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐻(y𝐺z)))
caovdird.2 (φA 𝑆)
caovdird.3 (φB 𝑆)
caovdird.4 (φ𝐶 𝐾)
Assertion
Ref Expression
caovdird (φ → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐻(B𝐺𝐶)))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   φ,x,y,z   x,𝐹,y,z   x,𝐺,y,z   x,𝐻,y,z   x,𝐾,y,z   x,𝑆,y,z

Proof of Theorem caovdird
StepHypRef Expression
1 id 19 . 2 (φφ)
2 caovdird.2 . 2 (φA 𝑆)
3 caovdird.3 . 2 (φB 𝑆)
4 caovdird.4 . 2 (φ𝐶 𝐾)
5 caovdirg.1 . . 3 ((φ (x 𝑆 y 𝑆 z 𝐾)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐻(y𝐺z)))
65caovdirg 5617 . 2 ((φ (A 𝑆 B 𝑆 𝐶 𝐾)) → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐻(B𝐺𝐶)))
71, 2, 3, 4, 6syl13anc 1136 1 (φ → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐻(B𝐺𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  (class class class)co 5452
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 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-iota 4809  df-fv 4852  df-ov 5455
This theorem is referenced by:  caovdilemd  5631  recexgt0sr  6653
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