ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifandc GIF version

Theorem ifandc 3478
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 805 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 iftrue 3449 . . . 4 (𝜑 → if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) = if(𝜓, 𝐴, 𝐵))
3 ibar 299 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
43ifbid 3463 . . . 4 (𝜑 → if(𝜓, 𝐴, 𝐵) = if((𝜑𝜓), 𝐴, 𝐵))
52, 4eqtr2d 2151 . . 3 (𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵))
6 simpl 108 . . . . . 6 ((𝜑𝜓) → 𝜑)
76con3i 606 . . . . 5 𝜑 → ¬ (𝜑𝜓))
87iffalsed 3454 . . . 4 𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = 𝐵)
9 iffalse 3452 . . . 4 𝜑 → if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) = 𝐵)
108, 9eqtr4d 2153 . . 3 𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵))
115, 10jaoi 690 . 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 682  DECID wdc 804   = wceq 1316  ifcif 3444
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-if 3445
This theorem is referenced by:  isumss  11128
  Copyright terms: Public domain W3C validator