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Theorem iscom 25436
 Description: The predicate "is a commutative operation". (Contributed by FL, 5-Sep-2010.)
Hypothesis
Ref Expression
iscom.1
Assertion
Ref Expression
iscom
Distinct variable group:   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem iscom
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 iscom.1 . . . . . 6
2 dmeq 4895 . . . . . . . 8
32eqcomd 2301 . . . . . . 7
43dmeqd 4897 . . . . . 6
51, 4syl5req 2341 . . . . 5
65eleq2d 2363 . . . 4
75eleq2d 2363 . . . . . 6
8 oveq 5880 . . . . . . 7
9 oveq 5880 . . . . . . 7
108, 9eqeq12d 2310 . . . . . 6
117, 10imbi12d 311 . . . . 5
1211ralbidv2 2578 . . . 4
136, 12imbi12d 311 . . 3
1413ralbidv2 2578 . 2
15 df-com1 25435 . 2
1614, 15elab2g 2929 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wceq 1632   wcel 1696  wral 2556   cdm 4705  (class class class)co 5874  ccm1 25434 This theorem is referenced by:  iscomb  25437  ablocomgrp  25445 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-dm 4715  df-iota 5235  df-fv 5279  df-ov 5877  df-com1 25435
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