Theorem ccased 1052

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 33	. . 3 ⊢ ((𝜓 ∧ 𝜒) → (𝜑 → 𝜂))
3		ccased.2	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜂))
4	3	com12 33	. . 3 ⊢ ((𝜃 ∧ 𝜒) → (𝜑 → 𝜂))
5		ccased.3	. . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜂))
6	5	com12 33	. . 3 ⊢ ((𝜓 ∧ 𝜏) → (𝜑 → 𝜂))
7		ccased.4	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜏) → 𝜂))
8	7	com12 33	. . 3 ⊢ ((𝜃 ∧ 𝜏) → (𝜑 → 𝜂))
9	2, 4, 6, 8	ccase 1051	. 2 ⊢ (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → (𝜑 → 𝜂))
10	9	com12 33	1 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → 𝜂))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860

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 210  df-an 401  df-or 861

This theorem is referenced by:  fvf1pr  7295  resf1extb  7919  fpwwe2lem12  10615  mulge0  11720  zmulcl  12634  lcmabs  16653  pospo  18389  mulgass  19168  indistopon  23119  lgsdir2lem5  27451  outsideofeq  36493  weiunpo  36838  smprngopr  38563  cdlemg33  41347  monotoddzzfi  43531  acongtr  43567  smprngprmrng  48959