Theorem 3jaodan 1248

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

Hypotheses

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

Assertion

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

Proof of Theorem 3jaodan

Step	Hyp	Ref	Expression
1		3jaodan.1	. . . 4 ⊢ ((φ ∧ ψ) → χ)
2	1	ex 423	. . 3 ⊢ (φ → (ψ → χ))
3		3jaodan.2	. . . 4 ⊢ ((φ ∧ θ) → χ)
4	3	ex 423	. . 3 ⊢ (φ → (θ → χ))
5		3jaodan.3	. . . 4 ⊢ ((φ ∧ τ) → χ)
6	5	ex 423	. . 3 ⊢ (φ → (τ → χ))
7	2, 4, 6	3jaod 1246	. 2 ⊢ (φ → ((ψ ∨ θ ∨ τ) → χ))
8	7	imp 418	1 ⊢ ((φ ∧ (ψ ∨ θ ∨ τ)) → χ)

Syntax hints:   → wi 4   ∧ wa 358   ∨ w3o 933

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: (None)