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Theorem iscom2 22000
 Description: A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
Assertion
Ref Expression
iscom2
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem iscom2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-com2 21999 . . . 4
21a1i 11 . . 3
32eleq2d 2503 . 2
4 rneq 5095 . . . 4
54raleqdv 2910 . . . 4
64, 5raleqbidv 2916 . . 3
7 oveq 6087 . . . . 5
8 oveq 6087 . . . . 5
97, 8eqeq12d 2450 . . . 4
1092ralbidv 2747 . . 3
116, 10opelopabg 4473 . 2
123, 11bitrd 245 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2705  cop 3817  copab 4265   crn 4879  (class class class)co 6081  ccm2 21998 This theorem is referenced by:  iscrngo2  26608  iscringd  26609 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889  df-iota 5418  df-fv 5462  df-ov 6084  df-com2 21999
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