Theorem ccased 1038

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 1037	. 2 ⊢ (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → (𝜑 → 𝜂))
10	9	com12 32	1 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → 𝜂))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847

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 207  df-an 396  df-or 848

This theorem is referenced by:  fvf1pr  7285  resf1extb  7913  fpwwe2lem12  10602  mulge0  11703  zmulcl  12589  lcmabs  16582  pospo  18311  mulgass  19050  indistopon  22895  lgsdir2lem5  27247  outsideofeq  36125  weiunpo  36460  smprngopr  38053  cdlemg33  40712  monotoddzzfi  42938  acongtr  42974