Theorem cadbi123d 1383

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

Hypotheses

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

Assertion

Ref	Expression
cadbi123d	⊢ (φ → (cadd(ψ, θ, η) ↔ cadd(χ, τ, ζ)))

Proof of Theorem cadbi123d

Step	Hyp	Ref	Expression
1		hadbid.1	. . . 4 ⊢ (φ → (ψ ↔ χ))
2		hadbid.2	. . . 4 ⊢ (φ → (θ ↔ τ))
3	1, 2	anbi12d 691	. . 3 ⊢ (φ → ((ψ ∧ θ) ↔ (χ ∧ τ)))
4		hadbid.3	. . . 4 ⊢ (φ → (η ↔ ζ))
5	1, 2	xorbi12d 1315	. . . 4 ⊢ (φ → ((ψ ⊻ θ) ↔ (χ ⊻ τ)))
6	4, 5	anbi12d 691	. . 3 ⊢ (φ → ((η ∧ (ψ ⊻ θ)) ↔ (ζ ∧ (χ ⊻ τ))))
7	3, 6	orbi12d 690	. 2 ⊢ (φ → (((ψ ∧ θ) ∨ (η ∧ (ψ ⊻ θ))) ↔ ((χ ∧ τ) ∨ (ζ ∧ (χ ⊻ τ)))))
8		df-cad 1381	. 2 ⊢ (cadd(ψ, θ, η) ↔ ((ψ ∧ θ) ∨ (η ∧ (ψ ⊻ θ))))
9		df-cad 1381	. 2 ⊢ (cadd(χ, τ, ζ) ↔ ((χ ∧ τ) ∨ (ζ ∧ (χ ⊻ τ))))
10	7, 8, 9	3bitr4g 279	1 ⊢ (φ → (cadd(ψ, θ, η) ↔ cadd(χ, τ, ζ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ⊻ wxo 1304  caddwcad 1379

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-xor 1305  df-cad 1381

This theorem is used by:  cadbi123i  1385