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

Theorem ifbothdc 3504
Description: A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
Hypotheses
Ref Expression
ifbothdc.1 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
ifbothdc.2 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
Assertion
Ref Expression
ifbothdc ((𝜓𝜒DECID 𝜑) → 𝜃)

Proof of Theorem ifbothdc
StepHypRef Expression
1 iftrue 3479 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21eqcomd 2145 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
3 ifbothdc.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
42, 3syl 14 . . . 4 (𝜑 → (𝜓𝜃))
54biimpcd 158 . . 3 (𝜓 → (𝜑𝜃))
653ad2ant1 1002 . 2 ((𝜓𝜒DECID 𝜑) → (𝜑𝜃))
7 iffalse 3482 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
87eqcomd 2145 . . . . 5 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
9 ifbothdc.2 . . . . 5 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
108, 9syl 14 . . . 4 𝜑 → (𝜒𝜃))
1110biimpcd 158 . . 3 (𝜒 → (¬ 𝜑𝜃))
12113ad2ant2 1003 . 2 ((𝜓𝜒DECID 𝜑) → (¬ 𝜑𝜃))
13 exmiddc 821 . . 3 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
14133ad2ant3 1004 . 2 ((𝜓𝜒DECID 𝜑) → (𝜑 ∨ ¬ 𝜑))
156, 12, 14mpjaod 707 1 ((𝜓𝜒DECID 𝜑) → 𝜃)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 697  DECID wdc 819  w3a 962   = wceq 1331  ifcif 3474
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-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-if 3475
This theorem is referenced by:  isumlessdc  11277
  Copyright terms: Public domain W3C validator