Theorem cadbi123i

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

	Ref	Expression
Hypotheses	cadbii.1	$$⊢\phi \leftrightarrow \psi$$
	cadbii.2	$$⊢\chi \leftrightarrow \theta$$
	cadbii.3	$$⊢\tau \leftrightarrow \eta$$
Assertion	cadbi123i	$$⊢cadd\left(\phi ,\chi ,\tau \right)\leftrightarrow cadd\left(\psi ,\theta ,\eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		cadbii.1	$$⊢\phi \leftrightarrow \psi$$
2		cadbii.2	$$⊢\chi \leftrightarrow \theta$$
3		cadbii.3	$$⊢\tau \leftrightarrow \eta$$
4	1	a1i	$$⊢\top \to \left(\phi \leftrightarrow \psi \right)$$
5	2	a1i	$$⊢\top \to \left(\chi \leftrightarrow \theta \right)$$
6	3	a1i	$$⊢\top \to \left(\tau \leftrightarrow \eta \right)$$
7	4 5 6	cadbi123d	$$⊢\top \to \left(cadd\left(\phi ,\chi ,\tau \right)\leftrightarrow cadd\left(\psi ,\theta ,\eta \right)\right)$$
8	7	mptru	$$⊢cadd\left(\phi ,\chi ,\tau \right)\leftrightarrow cadd\left(\psi ,\theta ,\eta \right)$$