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Theorem ifsbdc 3385
Description: Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)
Hypotheses
Ref Expression
ifsbdc.1 (if(𝜑, 𝐴, 𝐵) = 𝐴𝐶 = 𝐷)
ifsbdc.2 (if(𝜑, 𝐴, 𝐵) = 𝐵𝐶 = 𝐸)
Assertion
Ref Expression
ifsbdc (DECID 𝜑𝐶 = if(𝜑, 𝐷, 𝐸))

Proof of Theorem ifsbdc
StepHypRef Expression
1 exmiddc 778 . 2 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2 iftrue 3378 . . . . 5 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3 ifsbdc.1 . . . . 5 (if(𝜑, 𝐴, 𝐵) = 𝐴𝐶 = 𝐷)
42, 3syl 14 . . . 4 (𝜑𝐶 = 𝐷)
5 iftrue 3378 . . . 4 (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
64, 5eqtr4d 2118 . . 3 (𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
7 iffalse 3381 . . . . 5 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8 ifsbdc.2 . . . . 5 (if(𝜑, 𝐴, 𝐵) = 𝐵𝐶 = 𝐸)
97, 8syl 14 . . . 4 𝜑𝐶 = 𝐸)
10 iffalse 3381 . . . 4 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
119, 10eqtr4d 2118 . . 3 𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
126, 11jaoi 669 . 2 ((𝜑 ∨ ¬ 𝜑) → 𝐶 = if(𝜑, 𝐷, 𝐸))
131, 12syl 14 1 (DECID 𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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:  fvifdc  5272
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