Theorem 3jaoian 1247

Description: Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.)

Hypotheses

Ref	Expression
3jaoian.1	⊢ ((φ ∧ ψ) → χ)
3jaoian.2	⊢ ((θ ∧ ψ) → χ)
3jaoian.3	⊢ ((τ ∧ ψ) → χ)

Assertion

Ref	Expression
3jaoian	⊢ (((φ ∨ θ ∨ τ) ∧ ψ) → χ)

Proof of Theorem 3jaoian

Step	Hyp	Ref	Expression
1		3jaoian.1	. . . 4 ⊢ ((φ ∧ ψ) → χ)
2	1	ex 423	. . 3 ⊢ (φ → (ψ → χ))
3		3jaoian.2	. . . 4 ⊢ ((θ ∧ ψ) → χ)
4	3	ex 423	. . 3 ⊢ (θ → (ψ → χ))
5		3jaoian.3	. . . 4 ⊢ ((τ ∧ ψ) → χ)
6	5	ex 423	. . 3 ⊢ (τ → (ψ → χ))
7	2, 4, 6	3jaoi 1245	. 2 ⊢ ((φ ∨ θ ∨ τ) → (ψ → χ))
8	7	imp 418	1 ⊢ (((φ ∨ θ ∨ τ) ∧ ψ) → χ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∨ w3o 933

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-3or 935  df-3an 936

This theorem is used by: (None)