Theorem ccased 1044

Description: Deduction for combining cases. (Contributed by NM, 9-May-2004.)

Hypotheses

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

Assertion

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

Proof of Theorem ccased

Step	Hyp	Ref	Expression
1		ccased.1	. . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜂))
2	1	com12 32	. . 3 ⊢ ((𝜓 ∧ 𝜒) → (𝜑 → 𝜂))
3		ccased.2	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜂))
4	3	com12 32	. . 3 ⊢ ((𝜃 ∧ 𝜒) → (𝜑 → 𝜂))
5		ccased.3	. . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜂))
6	5	com12 32	. . 3 ⊢ ((𝜓 ∧ 𝜏) → (𝜑 → 𝜂))
7		ccased.4	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜏) → 𝜂))
8	7	com12 32	. . 3 ⊢ ((𝜃 ∧ 𝜏) → (𝜑 → 𝜂))
9	2, 4, 6, 8	ccase 1043	. 2 ⊢ (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → (𝜑 → 𝜂))
10	9	com12 32	1 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → 𝜂))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853

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 208  df-an 397  df-or 854

This theorem is referenced by:  fvf1pr  7258  resf1extb  7881  fpwwe2lem12  10563  mulge0  11666  zmulcl  12574  lcmabs  16572  pospo  18307  mulgass  19085  indistopon  22991  lgsdir2lem5  27317  outsideofeq  36365  weiunpo  36700  smprngopr  38426  cdlemg33  41210  monotoddzzfi  43394  acongtr  43430