Theorem wl-df2-3mintru2 35350

Description: The adder carry in disjunctive normal form. An alternative highly symmetric definition emphasizing the independence of order of the inputs 𝜑, 𝜓 and 𝜒. Copy of cador 1615. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 12-Jun-2024.)

Assertion

Ref	Expression
wl-df2-3mintru2	⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))

Proof of Theorem wl-df2-3mintru2

Step	Hyp	Ref	Expression
1		andi 1008	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
2	1	orbi1i 914	. 2 ⊢ (((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)))
3		wl-df-3mintru2 35349	. . 3 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)))
4		animorl 978	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜓 ∨ 𝜒))
5		wl-ifpimpr 35331	. . . 4 ⊢ (((𝜓 ∧ 𝜒) → (𝜓 ∨ 𝜒)) → (if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒))))
6	4, 5	ax-mp 5	. . 3 ⊢ (if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒)))
7	3, 6	bitri 278	. 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒)))
8		df-3or 1090	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)))
9	2, 7, 8	3bitr4i 306	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847  if-wif 1063   ∨ w3o 1088  caddwcad 1613

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 848  df-ifp 1064  df-3or 1090  df-3an 1091  df-xor 1508  df-cad 1614

This theorem is referenced by:  wl-df4-3mintru2  35352