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Theorem ifbi 3348
 Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
Assertion
Ref Expression
ifbi

Proof of Theorem ifbi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 anbi2 440 . . . 4
2 id 19 . . . . . 6
32notbid 592 . . . . 5
43anbi2d 437 . . . 4
51, 4orbi12d 707 . . 3
65abbidv 2155 . 2
7 df-if 3332 . 2
8 df-if 3332 . 2
96, 7, 83eqtr4g 2097 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 629   wceq 1243   wcel 1393  cab 2026  cif 3331 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-if 3332 This theorem is referenced by:  ifbid  3349  ifbieq2i  3351
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