Theorem wl-hadbi123i

Description: Equivalence theorem for the adder sum. Copy of hadbi123i . (Contributed by Mario Carneiro, 4-Sep-2016)

	Ref	Expression
Hypotheses	wl-hadbii.1	$$⊢\psi \leftrightarrow \chi$$
	wl-hadbii.2	$$⊢\theta \leftrightarrow \tau$$
	wl-hadbii.3	$$⊢\eta \leftrightarrow \zeta$$
Assertion	wl-hadbi123i	$$⊢hadd\left(\psi ,\theta ,\eta \right)\leftrightarrow hadd\left(\chi ,\tau ,\zeta \right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-hadbii.1	$$⊢\psi \leftrightarrow \chi$$
2		wl-hadbii.2	$$⊢\theta \leftrightarrow \tau$$
3		wl-hadbii.3	$$⊢\eta \leftrightarrow \zeta$$
4	1	a1i	$$⊢\top \to \left(\psi \leftrightarrow \chi \right)$$
5	2	a1i	$$⊢\top \to \left(\theta \leftrightarrow \tau \right)$$
6	3	a1i	$$⊢\top \to \left(\eta \leftrightarrow \zeta \right)$$
7	4 5 6	wl-hadbi123d	$$⊢\top \to \left(hadd\left(\psi ,\theta ,\eta \right)\leftrightarrow hadd\left(\chi ,\tau ,\zeta \right)\right)$$
8	7	mptru	$$⊢hadd\left(\psi ,\theta ,\eta \right)\leftrightarrow hadd\left(\chi ,\tau ,\zeta \right)$$