Theorem cadcoma 1645

Description: Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

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

Proof of Theorem cadcoma

Step	Hyp	Ref	Expression
1		ancom 466	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2		xorcom 1544	. . . 4 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))
3	2	anbi2i 635	. . 3 ⊢ ((𝜒 ∧ (𝜑 ⊻ 𝜓)) ↔ (𝜒 ∧ (𝜓 ⊻ 𝜑)))
4	1, 3	orbi12i 928	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))) ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ (𝜓 ⊻ 𝜑))))
5		df-cad 1640	. 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))
6		df-cad 1640	. 2 ⊢ (cadd(𝜓, 𝜑, 𝜒) ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ (𝜓 ⊻ 𝜑))))
7	4, 5, 6	3bitr4i 306	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜑, 𝜒))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861   ⊻ wxo 1541  caddwcad 1639

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-xor 1542  df-cad 1640

This theorem is used by:  cadrot  1647  sadcom  16546