Theorem wl-df4-3mintru2 37862

Description: An alternative definition of the adder carry. Copy of df-cad 1615. (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 1096	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))))
2		wl-df2-3mintru2 37860	. 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))
3		xor2 1525	. . . . . . 7 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
4	3	biancomi 464	. . . . . 6 ⊢ ((𝜑 ⊻ 𝜓) ↔ (¬ (𝜑 ∧ 𝜓) ∧ (𝜑 ∨ 𝜓)))
5	4	anbi1ci 633	. . . . 5 ⊢ ((𝜒 ∧ (𝜑 ⊻ 𝜓)) ↔ ((¬ (𝜑 ∧ 𝜓) ∧ (𝜑 ∨ 𝜓)) ∧ 𝜒))
6		anass 470	. . . . 5 ⊢ (((¬ (𝜑 ∧ 𝜓) ∧ (𝜑 ∨ 𝜓)) ∧ 𝜒) ↔ (¬ (𝜑 ∧ 𝜓) ∧ ((𝜑 ∨ 𝜓) ∧ 𝜒)))
7	5, 6	bitri 277	. . . 4 ⊢ ((𝜒 ∧ (𝜑 ⊻ 𝜓)) ↔ (¬ (𝜑 ∧ 𝜓) ∧ ((𝜑 ∨ 𝜓) ∧ 𝜒)))
8	7	orbi2i 919	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ (𝜑 ∧ 𝜓) ∧ ((𝜑 ∨ 𝜓) ∧ 𝜒))))
9		pm5.63 1028	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ ((𝜑 ∨ 𝜓) ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ (𝜑 ∧ 𝜓) ∧ ((𝜑 ∨ 𝜓) ∧ 𝜒))))
10		andir 1017	. . . 4 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))
11	10	orbi2i 919	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ ((𝜑 ∨ 𝜓) ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))))
12	8, 9, 11	3bitr2i 301	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))))
13	1, 2, 12	3bitr4i 305	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 397   ∨ wo 854   ∨ w3o 1092   ⊻ wxo 1519  caddwcad 1614

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 209  df-an 398  df-or 855  df-ifp 1070  df-3or 1094  df-3an 1095  df-xor 1520  df-cad 1615

This theorem is referenced by: (None)