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Theorem ifbothdadc 3402
Description: A formula 𝜃 containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.)
Hypotheses
Ref Expression
ifbothdc.1 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
ifbothdc.2 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
ifbothdadc.3 ((𝜂𝜑) → 𝜓)
ifbothdadc.4 ((𝜂 ∧ ¬ 𝜑) → 𝜒)
ifbothdadc.dc (𝜂DECID 𝜑)
Assertion
Ref Expression
ifbothdadc (𝜂𝜃)

Proof of Theorem ifbothdadc
StepHypRef Expression
1 ifbothdadc.3 . . 3 ((𝜂𝜑) → 𝜓)
2 iftrue 3378 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
32eqcomd 2088 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
4 ifbothdc.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
53, 4syl 14 . . . 4 (𝜑 → (𝜓𝜃))
65adantl 271 . . 3 ((𝜂𝜑) → (𝜓𝜃))
71, 6mpbid 145 . 2 ((𝜂𝜑) → 𝜃)
8 ifbothdadc.4 . . 3 ((𝜂 ∧ ¬ 𝜑) → 𝜒)
9 iffalse 3381 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
109eqcomd 2088 . . . . 5 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
11 ifbothdc.2 . . . . 5 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
1210, 11syl 14 . . . 4 𝜑 → (𝜒𝜃))
1312adantl 271 . . 3 ((𝜂 ∧ ¬ 𝜑) → (𝜒𝜃))
148, 13mpbid 145 . 2 ((𝜂 ∧ ¬ 𝜑) → 𝜃)
15 ifbothdadc.dc . . 3 (𝜂DECID 𝜑)
16 exmiddc 778 . . 3 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
1715, 16syl 14 . 2 (𝜂 → (𝜑 ∨ ¬ 𝜑))
187, 14, 17mpjaodan 745 1 (𝜂𝜃)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 776   = wceq 1285  ifcif 3373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-dc 777  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-if 3374
This theorem is referenced by:  hashgcdeq  10984
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