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Theorem wl-df4-3mintru2 34897
Description: An alternative definition of the adder carry. Copy of df-cad 1609. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 19-Jun-2024.)
Assertion
Ref Expression
wl-df4-3mintru2 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))))

Proof of Theorem wl-df4-3mintru2
StepHypRef Expression
1 3orass 1087 . 2 (((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ ((𝜑𝜒) ∨ (𝜓𝜒))))
2 wl-df2-3mintru2 34895 . 2 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)))
3 xor2 1509 . . . . . . 7 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
43biancomi 466 . . . . . 6 ((𝜑𝜓) ↔ (¬ (𝜑𝜓) ∧ (𝜑𝜓)))
54anbi1ci 628 . . . . 5 ((𝜒 ∧ (𝜑𝜓)) ↔ ((¬ (𝜑𝜓) ∧ (𝜑𝜓)) ∧ 𝜒))
6 anass 472 . . . . 5 (((¬ (𝜑𝜓) ∧ (𝜑𝜓)) ∧ 𝜒) ↔ (¬ (𝜑𝜓) ∧ ((𝜑𝜓) ∧ 𝜒)))
75, 6bitri 278 . . . 4 ((𝜒 ∧ (𝜑𝜓)) ↔ (¬ (𝜑𝜓) ∧ ((𝜑𝜓) ∧ 𝜒)))
87orbi2i 910 . . 3 (((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))) ↔ ((𝜑𝜓) ∨ (¬ (𝜑𝜓) ∧ ((𝜑𝜓) ∧ 𝜒))))
9 pm5.63 1017 . . 3 (((𝜑𝜓) ∨ ((𝜑𝜓) ∧ 𝜒)) ↔ ((𝜑𝜓) ∨ (¬ (𝜑𝜓) ∧ ((𝜑𝜓) ∧ 𝜒))))
10 andir 1006 . . . 4 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))
1110orbi2i 910 . . 3 (((𝜑𝜓) ∨ ((𝜑𝜓) ∧ 𝜒)) ↔ ((𝜑𝜓) ∨ ((𝜑𝜒) ∨ (𝜓𝜒))))
128, 9, 113bitr2i 302 . 2 (((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))) ↔ ((𝜑𝜓) ∨ ((𝜑𝜒) ∨ (𝜓𝜒))))
131, 2, 123bitr4i 306 1 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wo 844  w3o 1083  wxo 1502  caddwcad 1608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-xor 1503  df-cad 1609
This theorem is referenced by: (None)
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