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Theorem ifbieq12d2 23165
 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 404 . . . . 5
21a1i 10 . . . 4
3 ifbieq12d2.1 . . . . . . . . . 10
4 iftrue 3584 . . . . . . . . . 10
53, 4syl6bi 219 . . . . . . . . 9
65imp 418 . . . . . . . 8
7 ifbieq12d2.2 . . . . . . . 8
86, 7eqtr4d 2331 . . . . . . 7
98ex 423 . . . . . 6
109ancld 536 . . . . 5
113notbid 285 . . . . . . . . . 10
12 iffalse 3585 . . . . . . . . . 10
1311, 12syl6bi 219 . . . . . . . . 9
1413imp 418 . . . . . . . 8
15 ifbieq12d2.3 . . . . . . . 8
1614, 15eqtr4d 2331 . . . . . . 7
1716ex 423 . . . . . 6
1817ancld 536 . . . . 5
1910, 18orim12d 811 . . . 4
202, 19mpd 14 . . 3
21 eqif 3611 . . 3
2220, 21sylibr 203 . 2
2322eqcomd 2301 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wo 357   wa 358   wceq 1632  cif 3578 This theorem is referenced by:  itgeq12dv  23211 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-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-if 3579
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