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Theorem ifcomnan 4517
Description: Commute the conditions in two nested conditionals if both conditions are not simultaneously true. (Contributed by SO, 15-Jul-2018.)
Assertion
Ref Expression
ifcomnan (¬ (𝜑𝜓) → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)))

Proof of Theorem ifcomnan
StepHypRef Expression
1 pm3.13 988 . 2 (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
2 iffalse 4472 . . . 4 𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, 𝐶))
3 iffalse 4472 . . . . 5 𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐶)
43ifeq2d 4482 . . . 4 𝜑 → if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)) = if(𝜓, 𝐵, 𝐶))
52, 4eqtr4d 2856 . . 3 𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)))
6 iffalse 4472 . . . . 5 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶)
76ifeq2d 4482 . . . 4 𝜓 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜑, 𝐴, 𝐶))
8 iffalse 4472 . . . 4 𝜓 → if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)) = if(𝜑, 𝐴, 𝐶))
97, 8eqtr4d 2856 . . 3 𝜓 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)))
105, 9jaoi 851 . 2 ((¬ 𝜑 ∨ ¬ 𝜓) → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)))
111, 10syl 17 1 (¬ (𝜑𝜓) → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  ifcif 4463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-un 3938  df-if 4464
This theorem is referenced by:  mdetunilem6  21154
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