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Theorem ifandc 3513
 Description: Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
Assertion
Ref Expression
ifandc (DECID 𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵))

Proof of Theorem ifandc
StepHypRef Expression
1 df-dc 821 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 iftrue 3484 . . . 4 (𝜑 → if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) = if(𝜓, 𝐴, 𝐵))
3 ibar 299 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
43ifbid 3498 . . . 4 (𝜑 → if(𝜓, 𝐴, 𝐵) = if((𝜑𝜓), 𝐴, 𝐵))
52, 4eqtr2d 2174 . . 3 (𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵))
6 simpl 108 . . . . . 6 ((𝜑𝜓) → 𝜑)
76con3i 622 . . . . 5 𝜑 → ¬ (𝜑𝜓))
87iffalsed 3489 . . . 4 𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = 𝐵)
9 iffalse 3487 . . . 4 𝜑 → if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) = 𝐵)
108, 9eqtr4d 2176 . . 3 𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵))
115, 10jaoi 706 . 2 ((𝜑 ∨ ¬ 𝜑) → if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵))
121, 11sylbi 120 1 (DECID 𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 820   = wceq 1332  ifcif 3479 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-if 3480 This theorem is referenced by:  isumss  11193
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