Theorem 4casesdan 916

Description: Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.)

Hypotheses

Ref	Expression
4casesdan.1	⊢ ((φ ∧ (ψ ∧ χ)) → θ)
4casesdan.2	⊢ ((φ ∧ (ψ ∧ ¬ χ)) → θ)
4casesdan.3	⊢ ((φ ∧ (¬ ψ ∧ χ)) → θ)
4casesdan.4	⊢ ((φ ∧ (¬ ψ ∧ ¬ χ)) → θ)

Assertion

Ref	Expression
4casesdan	⊢ (φ → θ)

Proof of Theorem 4casesdan

Step	Hyp	Ref	Expression
1		4casesdan.1	. . 3 ⊢ ((φ ∧ (ψ ∧ χ)) → θ)
2	1	expcom 424	. 2 ⊢ ((ψ ∧ χ) → (φ → θ))
3		4casesdan.2	. . 3 ⊢ ((φ ∧ (ψ ∧ ¬ χ)) → θ)
4	3	expcom 424	. 2 ⊢ ((ψ ∧ ¬ χ) → (φ → θ))
5		4casesdan.3	. . 3 ⊢ ((φ ∧ (¬ ψ ∧ χ)) → θ)
6	5	expcom 424	. 2 ⊢ ((¬ ψ ∧ χ) → (φ → θ))
7		4casesdan.4	. . 3 ⊢ ((φ ∧ (¬ ψ ∧ ¬ χ)) → θ)
8	7	expcom 424	. 2 ⊢ ((¬ ψ ∧ ¬ χ) → (φ → θ))
9	2, 4, 6, 8	4cases 915	1 ⊢ (φ → θ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ 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-an 360

This theorem is referenced by: (None)