Theorem cad11 1399

Description: If two parameters are true, the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
cad11	⊢ ((φ ∧ ψ) → cadd(φ, ψ, χ))

Proof of Theorem cad11

Step	Hyp	Ref	Expression
1		orc 374	. 2 ⊢ ((φ ∧ ψ) → ((φ ∧ ψ) ∨ (χ ∧ (φ ⊻ ψ))))
2		df-cad 1381	. 2 ⊢ (cadd(φ, ψ, χ) ↔ ((φ ∧ ψ) ∨ (χ ∧ (φ ⊻ ψ))))
3	1, 2	sylibr 203	1 ⊢ ((φ ∧ ψ) → cadd(φ, ψ, χ))

Syntax hints:   → wi 4   ∨ 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-cad 1381

This theorem is referenced by:  cadtru  1401