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Theorem ifbothdadc 3498
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 3474 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
32eqcomd 2143 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐵))
4 ifbothdc.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))
53, 4syl 14 . . . 4 (𝜑 → (𝜓𝜃))
65adantl 275 . . 3 ((𝜂𝜑) → (𝜓𝜃))
71, 6mpbid 146 . 2 ((𝜂𝜑) → 𝜃)
8 ifbothdadc.4 . . 3 ((𝜂 ∧ ¬ 𝜑) → 𝜒)
9 iffalse 3477 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
109eqcomd 2143 . . . . 5 𝜑𝐵 = if(𝜑, 𝐴, 𝐵))
11 ifbothdc.2 . . . . 5 (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))
1210, 11syl 14 . . . 4 𝜑 → (𝜒𝜃))
1312adantl 275 . . 3 ((𝜂 ∧ ¬ 𝜑) → (𝜒𝜃))
148, 13mpbid 146 . 2 ((𝜂 ∧ ¬ 𝜑) → 𝜃)
15 ifbothdadc.dc . . 3 (𝜂DECID 𝜑)
16 exmiddc 821 . . 3 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
1715, 16syl 14 . 2 (𝜂 → (𝜑 ∨ ¬ 𝜑))
187, 14, 17mpjaodan 787 1 (𝜂𝜃)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  DECID wdc 819   = wceq 1331  ifcif 3469
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 2119
This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-if 3470
This theorem is referenced by:  hashgcdeq  11893
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