Theorem ccase 1053

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 971	. 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜓) → 𝜏)
4		ccase.3	. . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜏)
5		ccase.4	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜏)
6	4, 5	jaoian 971	. 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜃) → 𝜏)
7	3, 6	jaodan 972	1 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  ccased  1054  ccase2  1055  ssprsseq  4780  prel12g  4818  injresinjlem  13929  prodmo  16109  nn0rppwr  16741  nn0expgcd  16744  nn0gcdsq  16934  symgextf1  19641  cnmsgnsubg  21889  zseo  28814  dvdsexpnn0  43480  zaddcom  43622  zmulcom  43626  kelac2lem  44160  omcl3g  44430  usgrexmpl2trifr  49229