Theorem wl-df3-3mintru2 35168

Description: The adder carry in conjunctive normal form. An alternative highly symmetric definition emphasizing the independance of order of the inputs 𝜑, 𝜓 and 𝜒. Copy of cadan 1612. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 18-Jun-2024.)

Assertion

Ref	Expression
wl-df3-3mintru2	⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))

Proof of Theorem wl-df3-3mintru2

Step	Hyp	Ref	Expression
1		ordi 1004	. . 3 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
2	1	anbi1i 627	. 2 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜓 ∨ 𝜒)) ↔ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ (𝜓 ∨ 𝜒)))
3		wl-df-3mintru2 35166	. . 3 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)))
4		animorl 976	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜓 ∨ 𝜒))
5		wl-ifp4impr 35149	. . . 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-3an 1087	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) ∧ (𝜓 ∨ 𝜒)))
9	2, 7, 8	3bitr4i 307	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845  if-wif 1059   ∧ w3a 1085  caddwcad 1609

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 401  df-or 846  df-ifp 1060  df-3or 1086  df-3an 1087  df-xor 1504  df-cad 1610

This theorem is referenced by: (None)