Theorem ccase 977

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

Hypotheses

Ref	Expression
ccase.1	⊢ ((𝜑 ∧ 𝜓) → 𝜏)
ccase.2	⊢ ((𝜒 ∧ 𝜓) → 𝜏)
ccase.3	⊢ ((𝜑 ∧ 𝜃) → 𝜏)
ccase.4	⊢ ((𝜒 ∧ 𝜃) → 𝜏)

Assertion

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

Proof of Theorem ccase

Step	Hyp	Ref	Expression
1		ccase.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
2		ccase.2	. . 3 ⊢ ((𝜒 ∧ 𝜓) → 𝜏)
3	1, 2	jaoian 807	. 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜓) → 𝜏)
4		ccase.3	. . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜏)
5		ccase.4	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜏)
6	4, 5	jaoian 807	. 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜃) → 𝜏)
7	3, 6	jaodan 809	1 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  ccased  978  ccase2  979  undif3ss  3492  ssprsseq  3877  prodmodc  12364  nn0gcdsq  12999