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Theorem caov411d 5625
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovd.1 (φA 𝑆)
caovd.2 (φB 𝑆)
caovd.3 (φ𝐶 𝑆)
caovd.com ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
caovd.ass ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))
caovd.4 (φ𝐷 𝑆)
caovd.cl ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)
Assertion
Ref Expression
caov411d (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹B)𝐹(A𝐹𝐷)))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   x,𝐷,y,z   φ,x,y,z   x,𝐹,y,z   x,𝑆,y,z

Proof of Theorem caov411d
StepHypRef Expression
1 caovd.2 . . 3 (φB 𝑆)
2 caovd.1 . . 3 (φA 𝑆)
3 caovd.3 . . 3 (φ𝐶 𝑆)
4 caovd.com . . 3 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
5 caovd.ass . . 3 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))
6 caovd.4 . . 3 (φ𝐷 𝑆)
7 caovd.cl . . 3 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)
81, 2, 3, 4, 5, 6, 7caov4d 5624 . 2 (φ → ((B𝐹A)𝐹(𝐶𝐹𝐷)) = ((B𝐹𝐶)𝐹(A𝐹𝐷)))
94, 1, 2caovcomd 5596 . . 3 (φ → (B𝐹A) = (A𝐹B))
109oveq1d 5467 . 2 (φ → ((B𝐹A)𝐹(𝐶𝐹𝐷)) = ((A𝐹B)𝐹(𝐶𝐹𝐷)))
114, 1, 3caovcomd 5596 . . 3 (φ → (B𝐹𝐶) = (𝐶𝐹B))
1211oveq1d 5467 . 2 (φ → ((B𝐹𝐶)𝐹(A𝐹𝐷)) = ((𝐶𝐹B)𝐹(A𝐹𝐷)))
138, 10, 123eqtr3d 2077 1 (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹B)𝐹(A𝐹𝐷)))
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:  ecopovtrn  6132  ecopovtrng  6135  ltsonq  6375  ltanqg  6377  mulextsr1lem  6658
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