Theorem wl-hadbi123i 34769

Description: Equivalence theorem for the adder sum. Copy of hadbi123i 1596. (Contributed by Mario Carneiro, 4-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
wl-hadbi123i	⊢ (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁))

Proof of Theorem wl-hadbi123i

Step	Hyp	Ref	Expression
1		wl-hadbii.1	. . . 4 ⊢ (𝜓 ↔ 𝜒)
2	1	a1i 11	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
3		wl-hadbii.2	. . . 4 ⊢ (𝜃 ↔ 𝜏)
4	3	a1i 11	. . 3 ⊢ (⊤ → (𝜃 ↔ 𝜏))
5		wl-hadbii.3	. . . 4 ⊢ (𝜂 ↔ 𝜁)
6	5	a1i 11	. . 3 ⊢ (⊤ → (𝜂 ↔ 𝜁))
7	2, 4, 6	wl-hadbi123d 34768	. 2 ⊢ (⊤ → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))
8	7	mptru 1544	1 ⊢ (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁))

Syntax hints:   ↔ wb 208  ⊤wtru 1538  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: (None)