Metamath Proof Explorer

Theorem ccased

Description: Deduction for combining cases. (Contributed by NM, 9-May-2004)

Ref Expression
Hypotheses ccased.1 ⊢ φ → ψ ∧ χ → η
ccased.2 ⊢ φ → θ ∧ χ → η
ccased.3 ⊢ φ → ψ ∧ τ → η
ccased.4 ⊢ φ → θ ∧ τ → η
Assertion ccased ⊢ φ → ψ ∨ θ ∧ χ ∨ τ → η

Proof

Step Hyp Ref Expression
1 ccased.1 ⊢ φ → ψ ∧ χ → η
2 ccased.2 ⊢ φ → θ ∧ χ → η
3 ccased.3 ⊢ φ → ψ ∧ τ → η
4 ccased.4 ⊢ φ → θ ∧ τ → η
5 1 com12 ⊢ ψ ∧ χ → φ → η
6 2 com12 ⊢ θ ∧ χ → φ → η
7 3 com12 ⊢ ψ ∧ τ → φ → η
8 4 com12 ⊢ θ ∧ τ → φ → η
9 5 6 7 8 ccase ⊢ ψ ∨ θ ∧ χ ∨ τ → φ → η
10 9 com12 ⊢ φ → ψ ∨ θ ∧ χ ∨ τ → η