Theorem ccase2 907

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

Hypotheses

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

Assertion

Ref	Expression
ccase2	⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)

Proof of Theorem ccase2

Step	Hyp	Ref	Expression
1		ccase2.1	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
2		ccase2.2	. . 3 ⊢ (𝜒 → 𝜏)
3	2	adantr 270	. 2 ⊢ ((𝜒 ∧ 𝜓) → 𝜏)
4		ccase2.3	. . 3 ⊢ (𝜃 → 𝜏)
5	4	adantl 271	. 2 ⊢ ((𝜑 ∧ 𝜃) → 𝜏)
6	4	adantl 271	. 2 ⊢ ((𝜒 ∧ 𝜃) → 𝜏)
7	1, 3, 5, 6	ccase 905	1 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)