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Theorem ifbi 3342
Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
Assertion
Ref Expression
ifbi  if ,  ,  if ,  ,

Proof of Theorem ifbi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 anbi2 440 . . . 4
2 id 19 . . . . . 6
32notbid 591 . . . . 5
43anbi2d 437 . . . 4
51, 4orbi12d 706 . . 3
65abbidv 2152 . 2  {  |  }  {  |  }
7 df-if 3326 . 2  if ,  ,  {  |  }
8 df-if 3326 . 2  if ,  ,  {  |  }
96, 7, 83eqtr4g 2094 1  if ,  ,  if ,  ,
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 628   wceq 1242   wcel 1390   {cab 2023   ifcif 3325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-if 3326
This theorem is referenced by:  ifbid  3343  ifbieq2i  3345
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