Theorem jaoian 759

Description: Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)

Hypotheses

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

Assertion

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

Proof of Theorem jaoian

Step	Hyp	Ref	Expression
1		jaoian.1	. . . 4 ⊢ ((φ ∧ ψ) → χ)
2	1	ex 423	. . 3 ⊢ (φ → (ψ → χ))
3		jaoian.2	. . . 4 ⊢ ((θ ∧ ψ) → χ)
4	3	ex 423	. . 3 ⊢ (θ → (ψ → χ))
5	2, 4	jaoi 368	. 2 ⊢ ((φ ∨ θ) → (ψ → χ))
6	5	imp 418	1 ⊢ (((φ ∨ θ) ∧ ψ) → χ)

Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358

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

This theorem is referenced by:  ccase  912