Theorem wl-df2-3mintru2 35635

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 1613. (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 1004	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
2	1	orbi1i 910	. 2 ⊢ (((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)))
3		wl-df-3mintru2 35634	. . 3 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)))
4		animorl 974	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜓 ∨ 𝜒))
5		wl-ifpimpr 35616	. . . 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 274	. 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ∨ (𝜓 ∧ 𝜒)))
8		df-3or 1086	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)))
9	2, 7, 8	3bitr4i 302	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843  if-wif 1059   ∨ w3o 1084  caddwcad 1611

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 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-xor 1506  df-cad 1612

This theorem is referenced by:  wl-df4-3mintru2  35637