Theorem hadbi123d 1625

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

Hypotheses

Ref	Expression
hadbid.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
hadbid.2	⊢ (𝜑 → (𝜃 ↔ 𝜏))
hadbid.3	⊢ (𝜑 → (𝜂 ↔ 𝜁))

Assertion

Ref	Expression
hadbi123d	⊢ (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))

Proof of Theorem hadbi123d

Step	Hyp	Ref	Expression
1		hadbid.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		hadbid.2	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
3	1, 2	xorbi12d 1555	. . 3 ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))
4		hadbid.3	. . 3 ⊢ (𝜑 → (𝜂 ↔ 𝜁))
5	3, 4	xorbi12d 1555	. 2 ⊢ (𝜑 → (((𝜓 ⊻ 𝜃) ⊻ 𝜂) ↔ ((𝜒 ⊻ 𝜏) ⊻ 𝜁)))
6		df-had 1624	. 2 ⊢ (hadd(𝜓, 𝜃, 𝜂) ↔ ((𝜓 ⊻ 𝜃) ⊻ 𝜂))
7		df-had 1624	. 2 ⊢ (hadd(𝜒, 𝜏, 𝜁) ↔ ((𝜒 ⊻ 𝜏) ⊻ 𝜁))
8	5, 6, 7	3bitr4g 317	1 ⊢ (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ⊻ wxo 1541  haddwhad 1623

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-xor 1542  df-had 1624

This theorem is used by:  hadbi123i  1626  sadfval  16589  sadval  16593