Theorem cadbi123i 1385

Description: Equality theorem for adder carry. (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
cadbi123i	⊢ (cadd(φ, χ, τ) ↔ cadd(ψ, θ, η))

Proof of Theorem cadbi123i

Step	Hyp	Ref	Expression
1		hadbii.1	. . . 4 ⊢ (φ ↔ ψ)
2	1	a1i 10	. . 3 ⊢ ( ⊤ → (φ ↔ ψ))
3		hadbii.2	. . . 4 ⊢ (χ ↔ θ)
4	3	a1i 10	. . 3 ⊢ ( ⊤ → (χ ↔ θ))
5		hadbii.3	. . . 4 ⊢ (τ ↔ η)
6	5	a1i 10	. . 3 ⊢ ( ⊤ → (τ ↔ η))
7	2, 4, 6	cadbi123d 1383	. 2 ⊢ ( ⊤ → (cadd(φ, χ, τ) ↔ cadd(ψ, θ, η)))
8	7	trud 1323	1 ⊢ (cadd(φ, χ, τ) ↔ cadd(ψ, θ, η))

Syntax hints:   ↔ wb 176   ⊤ wtru 1316  caddwcad 1379

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 177  df-or 359  df-an 360  df-xor 1305  df-tru 1319  df-cad 1381

This theorem is referenced by: (None)