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Metamath Proof Explorer

Theorem ccase

Description: Inference for combining cases. (Contributed by NM, 29-Jul-1999) (Proof shortened by Wolf Lammen, 6-Jan-2013)

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

Proof

Step Hyp Ref Expression
1 ccase.1 ⊢ φ ∧ ψ → τ
2 ccase.2 ⊢ χ ∧ ψ → τ
3 ccase.3 ⊢ φ ∧ θ → τ
4 ccase.4 ⊢ χ ∧ θ → τ
5 1 2 jaoian ⊢ φ ∨ χ ∧ ψ → τ
6 3 4 jaoian ⊢ φ ∨ χ ∧ θ → τ
7 5 6 jaodan ⊢ φ ∨ χ ∧ ψ ∨ θ → τ