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Theorem ifbieq12d2 28161
 Description: Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.)
Hypotheses
Ref Expression
ifbieq12d2.1
ifbieq12d2.2
ifbieq12d2.3
Assertion
Ref Expression
ifbieq12d2

Proof of Theorem ifbieq12d2
StepHypRef Expression
1 exmid 416 . . . 4
2 ifbieq12d2.1 . . . . . . . . . 10
3 iftrue 3917 . . . . . . . . . 10
42, 3syl6bi 231 . . . . . . . . 9
54imp 430 . . . . . . . 8
6 ifbieq12d2.2 . . . . . . . 8
75, 6eqtr4d 2466 . . . . . . 7
87ex 435 . . . . . 6
98ancld 555 . . . . 5
102notbid 295 . . . . . . . . . 10
11 iffalse 3920 . . . . . . . . . 10
1210, 11syl6bi 231 . . . . . . . . 9
1312imp 430 . . . . . . . 8
14 ifbieq12d2.3 . . . . . . . 8
1513, 14eqtr4d 2466 . . . . . . 7
1615ex 435 . . . . . 6
1716ancld 555 . . . . 5
189, 17orim12d 846 . . . 4
191, 18mpi 20 . . 3
20 eqif 3949 . . 3
2119, 20sylibr 215 . 2
2221eqcomd 2430 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wo 369   wa 370   wceq 1437  cif 3911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-if 3912 This theorem is referenced by:  itgeq12dv  29167  sgnneg  29419  ofccat  29437
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