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Metamath Proof Explorer

Theorem ccase2

Description: Inference for combining cases. (Contributed by NM, 29-Jul-1999)

Ref Expression
Hypotheses ccase2.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜏 )
ccase2.2 ⊢ ( 𝜒 → 𝜏 )
ccase2.3 ⊢ ( 𝜃 → 𝜏 )
Assertion ccase2 ( ( ( 𝜑 ∨ 𝜒 ) ∧ ( 𝜓 ∨ 𝜃 ) ) → 𝜏 )

Proof

Step Hyp Ref Expression
1 ccase2.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜏 )
2 ccase2.2 ⊢ ( 𝜒 → 𝜏 )
3 ccase2.3 ⊢ ( 𝜃 → 𝜏 )
4 2 adantr ⊢ ( ( 𝜒 ∧ 𝜓 ) → 𝜏 )
5 3 adantl ⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
6 3 adantl ⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜏 )
7 1 4 5 6 ccase ⊢ ( ( ( 𝜑 ∨ 𝜒 ) ∧ ( 𝜓 ∨ 𝜃 ) ) → 𝜏 )