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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 )