Theorem cadcomb 1396

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

Assertion

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

Proof of Theorem cadcomb

Step	Hyp	Ref	Expression
1		3orcoma 942	. . 3 ⊢ (((φ ∧ ψ) ∨ (φ ∧ χ) ∨ (ψ ∧ χ)) ↔ ((φ ∧ χ) ∨ (φ ∧ ψ) ∨ (ψ ∧ χ)))
2		biid 227	. . . 4 ⊢ ((φ ∧ χ) ↔ (φ ∧ χ))
3		biid 227	. . . 4 ⊢ ((φ ∧ ψ) ↔ (φ ∧ ψ))
4		ancom 437	. . . 4 ⊢ ((ψ ∧ χ) ↔ (χ ∧ ψ))
5	2, 3, 4	3orbi123i 1141	. . 3 ⊢ (((φ ∧ χ) ∨ (φ ∧ ψ) ∨ (ψ ∧ χ)) ↔ ((φ ∧ χ) ∨ (φ ∧ ψ) ∨ (χ ∧ ψ)))
6	1, 5	bitri 240	. 2 ⊢ (((φ ∧ ψ) ∨ (φ ∧ χ) ∨ (ψ ∧ χ)) ↔ ((φ ∧ χ) ∨ (φ ∧ ψ) ∨ (χ ∧ ψ)))
7		cador 1391	. 2 ⊢ (cadd(φ, ψ, χ) ↔ ((φ ∧ ψ) ∨ (φ ∧ χ) ∨ (ψ ∧ χ)))
8		cador 1391	. 2 ⊢ (cadd(φ, χ, ψ) ↔ ((φ ∧ χ) ∨ (φ ∧ ψ) ∨ (χ ∧ ψ)))
9	6, 7, 8	3bitr4i 268	1 ⊢ (cadd(φ, ψ, χ) ↔ cadd(φ, χ, ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∨ w3o 933  caddwcad 1379

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 177  df-or 359  df-an 360  df-3or 935  df-xor 1305  df-cad 1381

This theorem is referenced by:  cadrot  1397