Theorem wl-df4-3mintru2 35658

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) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))

Proof of Theorem wl-df4-3mintru2

Step	Hyp	Ref	Expression
1		3orass 1089	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))))
2		wl-df2-3mintru2 35656	. 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))
3		xor2 1513	. . . . . . 7 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
4	3	biancomi 463	. . . . . 6 ⊢ ((𝜑 ⊻ 𝜓) ↔ (¬ (𝜑 ∧ 𝜓) ∧ (𝜑 ∨ 𝜓)))
5	4	anbi1ci 626	. . . . 5 ⊢ ((𝜒 ∧ (𝜑 ⊻ 𝜓)) ↔ ((¬ (𝜑 ∧ 𝜓) ∧ (𝜑 ∨ 𝜓)) ∧ 𝜒))
6		anass 469	. . . . 5 ⊢ (((¬ (𝜑 ∧ 𝜓) ∧ (𝜑 ∨ 𝜓)) ∧ 𝜒) ↔ (¬ (𝜑 ∧ 𝜓) ∧ ((𝜑 ∨ 𝜓) ∧ 𝜒)))
7	5, 6	bitri 274	. . . 4 ⊢ ((𝜒 ∧ (𝜑 ⊻ 𝜓)) ↔ (¬ (𝜑 ∧ 𝜓) ∧ ((𝜑 ∨ 𝜓) ∧ 𝜒)))
8	7	orbi2i 910	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ (𝜑 ∧ 𝜓) ∧ ((𝜑 ∨ 𝜓) ∧ 𝜒))))
9		pm5.63 1017	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ ((𝜑 ∨ 𝜓) ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ (𝜑 ∧ 𝜓) ∧ ((𝜑 ∨ 𝜓) ∧ 𝜒))))
10		andir 1006	. . . 4 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))
11	10	orbi2i 910	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ ((𝜑 ∨ 𝜓) ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))))
12	8, 9, 11	3bitr2i 299	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))))
13	1, 2, 12	3bitr4i 303	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∨ w3o 1085   ⊻ wxo 1506  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 206  df-an 397  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-xor 1507  df-cad 1609

This theorem is referenced by: (None)