Theorem jaodan 760

Description: Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)

Hypotheses

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

Assertion

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

Proof of Theorem jaodan

Step	Hyp	Ref	Expression
1		jaodan.1	. . . 4 ⊢ ((φ ∧ ψ) → χ)
2	1	ex 423	. . 3 ⊢ (φ → (ψ → χ))
3		jaodan.2	. . . 4 ⊢ ((φ ∧ θ) → χ)
4	3	ex 423	. . 3 ⊢ (φ → (θ → χ))
5	2, 4	jaod 369	. 2 ⊢ (φ → ((ψ ∨ θ) → χ))
6	5	imp 418	1 ⊢ ((φ ∧ (ψ ∨ θ)) → 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:  mpjaodan  761  andi  837  ccase  912