Theorem ccase 1022

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 941	. 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜓) → 𝜏)
4		ccase.3	. . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜏)
5		ccase.4	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜏)
6	4, 5	jaoian 941	. 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜃) → 𝜏)
7	3, 6	jaodan 942	1 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 382   ∨ wo 836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837

This theorem is referenced by:  ccased  1023  ccase2  1024  ssprsseq  4492  prel12g  4531  injresinjlem  12796  prodmo  14873  nn0gcdsq  15667  symgextf1  18048  cnmsgnsubg  20138  kelac2lem  38160