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Theorem ifsbdc 3491
 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 822 . 2 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2 iftrue 3484 . . . . 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 3484 . . . 4 (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
64, 5eqtr4d 2176 . . 3 (𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
7 iffalse 3487 . . . . 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 3487 . . . 4 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
119, 10eqtr4d 2176 . . 3 𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
126, 11jaoi 706 . 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 698  DECID wdc 820   = wceq 1332  ifcif 3479 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-dc 821  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-if 3480 This theorem is referenced by:  fvifdc  5451
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