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 )