Theorem cadbi123i

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

	Ref	Expression
Hypotheses	cadbii.1	⊢ ( 𝜑 ↔ 𝜓 )
	cadbii.2	⊢ ( 𝜒 ↔ 𝜃 )
	cadbii.3	⊢ ( 𝜏 ↔ 𝜂 )
Assertion	cadbi123i	⊢ ( cadd ( 𝜑 , 𝜒 , 𝜏 ) ↔ cadd ( 𝜓 , 𝜃 , 𝜂 ) )

Proof

Step	Hyp	Ref	Expression
1		cadbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2		cadbii.2	⊢ ( 𝜒 ↔ 𝜃 )
3		cadbii.3	⊢ ( 𝜏 ↔ 𝜂 )
4	1	a1i	⊢ ( ⊤ → ( 𝜑 ↔ 𝜓 ) )
5	2	a1i	⊢ ( ⊤ → ( 𝜒 ↔ 𝜃 ) )
6	3	a1i	⊢ ( ⊤ → ( 𝜏 ↔ 𝜂 ) )
7	4 5 6	cadbi123d	⊢ ( ⊤ → ( cadd ( 𝜑 , 𝜒 , 𝜏 ) ↔ cadd ( 𝜓 , 𝜃 , 𝜂 ) ) )
8	7	mptru	⊢ ( cadd ( 𝜑 , 𝜒 , 𝜏 ) ↔ cadd ( 𝜓 , 𝜃 , 𝜂 ) )