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Theorem ifbieq12d2 24003
 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 406 . . . 4
2 ifbieq12d2.1 . . . . . . . . . 10
3 iftrue 3746 . . . . . . . . . 10
42, 3syl6bi 221 . . . . . . . . 9
54imp 420 . . . . . . . 8
6 ifbieq12d2.2 . . . . . . . 8
75, 6eqtr4d 2472 . . . . . . 7
87ex 425 . . . . . 6
98ancld 538 . . . . 5
102notbid 287 . . . . . . . . . 10
11 iffalse 3747 . . . . . . . . . 10
1210, 11syl6bi 221 . . . . . . . . 9
1312imp 420 . . . . . . . 8
14 ifbieq12d2.3 . . . . . . . 8
1513, 14eqtr4d 2472 . . . . . . 7
1615ex 425 . . . . . 6
1716ancld 538 . . . . 5
189, 17orim12d 813 . . . 4
191, 18mpi 17 . . 3
20 eqif 3773 . . 3
2119, 20sylibr 205 . 2
2221eqcomd 2442 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wo 359   wa 360   wceq 1653  cif 3740 This theorem is referenced by:  itgeq12dv  24642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-if 3741
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