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Theorem ifbothdadc 3549
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 3523 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
32eqcomd 2170 . . . . 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 3526 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
109eqcomd 2170 . . . . 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 826 . . 3 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
1715, 16syl 14 . 2 (𝜂 → (𝜑 ∨ ¬ 𝜑))
187, 14, 17mpjaodan 788 1 (𝜂𝜃)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 824   = wceq 1342  ifcif 3518
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 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-dc 825  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-if 3519
This theorem is referenced by:  hashgcdeq  12165
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