Theorem pm5.53 771

Description: Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm5.53	⊢ ((((φ ∨ ψ) ∨ χ) → θ) ↔ (((φ → θ) ∧ (ψ → θ)) ∧ (χ → θ)))

Proof of Theorem pm5.53

Step	Hyp	Ref	Expression
1		jaob 758	. 2 ⊢ ((((φ ∨ ψ) ∨ χ) → θ) ↔ (((φ ∨ ψ) → θ) ∧ (χ → θ)))
2		jaob 758	. . 3 ⊢ (((φ ∨ ψ) → θ) ↔ ((φ → θ) ∧ (ψ → θ)))
3	2	anbi1i 676	. 2 ⊢ ((((φ ∨ ψ) → θ) ∧ (χ → θ)) ↔ (((φ → θ) ∧ (ψ → θ)) ∧ (χ → θ)))
4	1, 3	bitri 240	1 ⊢ ((((φ ∨ ψ) ∨ χ) → θ) ↔ (((φ → θ) ∧ (ψ → θ)) ∧ (χ → θ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

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

This theorem is used by: (None)