Theorem cadcomb 1616

Description: Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)

Assertion

Ref	Expression
cadcomb	⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜑, 𝜒, 𝜓))

Proof of Theorem cadcomb

Step	Hyp	Ref	Expression
1		cadan 1612	. . 3 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))
2		3ancoma 1096	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜓 ∨ 𝜒)))
3		orcom 866	. . . 4 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓))
4	3	3anbi3i 1157	. . 3 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜓)))
5	1, 2, 4	3bitri 296	. 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜓)))
6		cadan 1612	. 2 ⊢ (cadd(𝜑, 𝜒, 𝜓) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜓)))
7	5, 6	bitr4i 277	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜑, 𝜒, 𝜓))

Syntax hints:   ↔ wb 205   ∨ wo 843   ∧ 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 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-xor 1504  df-cad 1610

This theorem is referenced by:  cadrot  1617