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Theorem caov12d 5624
 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)))
Assertion
Ref Expression
caov12d (φ → (A𝐹(B𝐹𝐶)) = (B𝐹(A𝐹𝐶)))
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

Proof of Theorem caov12d
StepHypRef Expression
1 caovd.com . . . 4 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
2 caovd.1 . . . 4 (φA 𝑆)
3 caovd.2 . . . 4 (φB 𝑆)
41, 2, 3caovcomd 5599 . . 3 (φ → (A𝐹B) = (B𝐹A))
54oveq1d 5470 . 2 (φ → ((A𝐹B)𝐹𝐶) = ((B𝐹A)𝐹𝐶))
6 caovd.ass . . 3 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))
7 caovd.3 . . 3 (φ𝐶 𝑆)
86, 2, 3, 7caovassd 5602 . 2 (φ → ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶)))
96, 3, 2, 7caovassd 5602 . 2 (φ → ((B𝐹A)𝐹𝐶) = (B𝐹(A𝐹𝐶)))
105, 8, 93eqtr3d 2077 1 (φ → (A𝐹(B𝐹𝐶)) = (B𝐹(A𝐹𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ 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:  caov4d  5627  caovimo  5636  ltaddnq  6390  ltexnqq  6391  enq0tr  6416  mullocprlem  6550  1idprl  6565  1idpru  6566  cauappcvgprlemdisj  6622  mulcmpblnrlemg  6648  lttrsr  6670  ltsosr  6672  0idsr  6675  1idsr  6676  recexgt0sr  6681  mulgt0sr  6684  axmulass  6737
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