Theorem wl-hadbi123d 34768

Description: Equivalence theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024.)

Hypotheses

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

Assertion

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

Proof of Theorem wl-hadbi123d

Step	Hyp	Ref	Expression
1		wl-hadbid.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		wl-hadbid.2	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
3	1, 2	bibi12d 348	. . 3 ⊢ (𝜑 → ((𝜓 ↔ 𝜃) ↔ (𝜒 ↔ 𝜏)))
4		wl-hadbid.3	. . 3 ⊢ (𝜑 → (𝜂 ↔ 𝜁))
5	3, 4	bibi12d 348	. 2 ⊢ (𝜑 → (((𝜓 ↔ 𝜃) ↔ 𝜂) ↔ ((𝜒 ↔ 𝜏) ↔ 𝜁)))
6		wl-dfhad3 34765	. 2 ⊢ (hadd(𝜓, 𝜃, 𝜂) ↔ ((𝜓 ↔ 𝜃) ↔ 𝜂))
7		wl-dfhad3 34765	. 2 ⊢ (hadd(𝜒, 𝜏, 𝜁) ↔ ((𝜒 ↔ 𝜏) ↔ 𝜁))
8	5, 6, 7	3bitr4g 316	1 ⊢ (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))

Syntax hints:   → wi 4   ↔ wb 208  haddwhad 1593

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 209  df-an 399  df-or 844  df-ifp 1058  df-xor 1502  df-tru 1540  df-had 1594

This theorem is referenced by:  wl-hadbi123i  34769