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Theorem com2i 25519
 Description: Property of a commutative structure with two operation. (Contributed by FL, 14-Feb-2010.)
Hypotheses
Ref Expression
com2i.1
com2i.2
com2i.3
Assertion
Ref Expression
com2i
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem com2i
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 com2i.1 . . . . . 6
21eqcomi 2300 . . . . 5
32eqeq2i 2306 . . . 4
4 rneq 4920 . . . . . 6
5 com2i.3 . . . . . 6
64, 5syl6eqr 2346 . . . . 5
7 raleq 2749 . . . . . 6
87raleqbi1dv 2757 . . . . 5
96, 8syl 15 . . . 4
103, 9sylbi 187 . . 3
11 com2i.2 . . . . . . 7
1211eqcomi 2300 . . . . . 6
1312eqeq2i 2306 . . . . 5
14 oveq 5880 . . . . . 6
15 oveq 5880 . . . . . 6
1614, 15eqeq12d 2310 . . . . 5
1713, 16sylbi 187 . . . 4
18172ralbidv 2598 . . 3
1910, 18elopabi 6201 . 2
20 df-com2 21094 . 2
2119, 20eleq2s 2388 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wceq 1632   wcel 1696  wral 2556  copab 4092   crn 4706  cfv 5271  (class class class)co 5874  c1st 6136  c2nd 6137  ccm2 21093 This theorem is referenced by:  fldi  25530 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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-com2 21094
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