Theorem cadbi123d 1613

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

Hypotheses

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

Assertion

Ref	Expression
cadbi123d	⊢ (𝜑 → (cadd(𝜓, 𝜃, 𝜂) ↔ cadd(𝜒, 𝜏, 𝜁)))

Proof of Theorem cadbi123d

Step	Hyp	Ref	Expression
1		cadbid.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		cadbid.2	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
3	1, 2	anbi12d 630	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜏)))
4		cadbid.3	. . . 4 ⊢ (𝜑 → (𝜂 ↔ 𝜁))
5	1, 2	xorbi12d 1519	. . . 4 ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))
6	4, 5	anbi12d 630	. . 3 ⊢ (𝜑 → ((𝜂 ∧ (𝜓 ⊻ 𝜃)) ↔ (𝜁 ∧ (𝜒 ⊻ 𝜏))))
7	3, 6	orbi12d 915	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜃) ∨ (𝜂 ∧ (𝜓 ⊻ 𝜃))) ↔ ((𝜒 ∧ 𝜏) ∨ (𝜁 ∧ (𝜒 ⊻ 𝜏)))))
8		df-cad 1610	. 2 ⊢ (cadd(𝜓, 𝜃, 𝜂) ↔ ((𝜓 ∧ 𝜃) ∨ (𝜂 ∧ (𝜓 ⊻ 𝜃))))
9		df-cad 1610	. 2 ⊢ (cadd(𝜒, 𝜏, 𝜁) ↔ ((𝜒 ∧ 𝜏) ∨ (𝜁 ∧ (𝜒 ⊻ 𝜏))))
10	7, 8, 9	3bitr4g 313	1 ⊢ (𝜑 → (cadd(𝜓, 𝜃, 𝜂) ↔ cadd(𝜒, 𝜏, 𝜁)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ⊻ wxo 1503  caddwcad 1609

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 206  df-an 396  df-or 844  df-xor 1504  df-cad 1610

This theorem is referenced by:  cadbi123i  1614  sadfval  16087  sadcp1  16090