Theorem hadbi123i 1597

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

Hypotheses

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

Assertion

Ref	Expression
hadbi123i	⊢ (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂))

Proof of Theorem hadbi123i

Step	Hyp	Ref	Expression
1		hadbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	a1i 11	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3		hadbii.2	. . . 4 ⊢ (𝜒 ↔ 𝜃)
4	3	a1i 11	. . 3 ⊢ (⊤ → (𝜒 ↔ 𝜃))
5		hadbii.3	. . . 4 ⊢ (𝜏 ↔ 𝜂)
6	5	a1i 11	. . 3 ⊢ (⊤ → (𝜏 ↔ 𝜂))
7	2, 4, 6	hadbi123d 1596	. 2 ⊢ (⊤ → (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂)))
8	7	mptru 1546	1 ⊢ (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂))

Syntax hints:   ↔ wb 205  ⊤wtru 1540  haddwhad 1594

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-xor 1507  df-tru 1542  df-had 1595

This theorem is referenced by: (None)