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Theorem cadcoma 1395
Description: Commutative law for 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
StepHypRef Expression
1 ancom 437 . . 3 ((φ ψ) ↔ (ψ φ))
2 xorcom 1307 . . . 4 ((φψ) ↔ (ψφ))
32anbi2i 675 . . 3 ((χ (φψ)) ↔ (χ (ψφ)))
41, 3orbi12i 507 . 2 (((φ ψ) (χ (φψ))) ↔ ((ψ φ) (χ (ψφ))))
5 df-cad 1381 . 2 (cadd(φ, ψ, χ) ↔ ((φ ψ) (χ (φψ))))
6 df-cad 1381 . 2 (cadd(ψ, φ, χ) ↔ ((ψ φ) (χ (ψφ))))
74, 5, 63bitr4i 268 1 (cadd(φ, ψ, χ) ↔ cadd(ψ, φ, χ))
Colors of variables: wff setvar class
Syntax hints:  wb 176   wo 357   wa 358  wxo 1304  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-xor 1305  df-cad 1381
This theorem is referenced by:  cadrot  1397
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