Theorem cadbi123i 1613

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

Hypotheses

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

Assertion

Ref	Expression
cadbi123i	⊢ (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂))

Proof of Theorem cadbi123i

Step	Hyp	Ref	Expression
1		cadbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	a1i 11	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3		cadbii.2	. . . 4 ⊢ (𝜒 ↔ 𝜃)
4	3	a1i 11	. . 3 ⊢ (⊤ → (𝜒 ↔ 𝜃))
5		cadbii.3	. . . 4 ⊢ (𝜏 ↔ 𝜂)
6	5	a1i 11	. . 3 ⊢ (⊤ → (𝜏 ↔ 𝜂))
7	2, 4, 6	cadbi123d 1612	. 2 ⊢ (⊤ → (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂)))
8	7	mptru 1546	1 ⊢ (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂))

Syntax hints:   ↔ wb 205  ⊤wtru 1540  caddwcad 1608

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 397  df-or 845  df-xor 1507  df-tru 1542  df-cad 1609

This theorem is referenced by: (None)