Theorem 3jaoi 1245

Description: Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)

Hypotheses

Ref	Expression
3jaoi.1	⊢ (φ → ψ)
3jaoi.2	⊢ (χ → ψ)
3jaoi.3	⊢ (θ → ψ)

Assertion

Ref	Expression
3jaoi	⊢ ((φ ∨ χ ∨ θ) → ψ)

Proof of Theorem 3jaoi

Step	Hyp	Ref	Expression
1		3jaoi.1	. . 3 ⊢ (φ → ψ)
2		3jaoi.2	. . 3 ⊢ (χ → ψ)
3		3jaoi.3	. . 3 ⊢ (θ → ψ)
4	1, 2, 3	3pm3.2i 1130	. 2 ⊢ ((φ → ψ) ∧ (χ → ψ) ∧ (θ → ψ))
5		3jao 1243	. 2 ⊢ (((φ → ψ) ∧ (χ → ψ) ∧ (θ → ψ)) → ((φ ∨ χ ∨ θ) → ψ))
6	4, 5	ax-mp 5	1 ⊢ ((φ ∨ χ ∨ θ) → ψ)

Syntax hints:   → wi 4   ∨ w3o 933   ∧ w3a 934

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-3or 935  df-3an 936

This theorem is referenced by:  3jaoian  1247