Theorem ccased 1039

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 1038	. 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 848

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 849

This theorem is referenced by:  fvf1pr  7327  resf1extb  7956  fpwwe2lem12  10682  mulge0  11781  zmulcl  12666  lcmabs  16642  pospo  18390  mulgass  19129  indistopon  23008  lgsdir2lem5  27373  outsideofeq  36131  weiunpo  36466  smprngopr  38059  cdlemg33  40713  monotoddzzfi  42954  acongtr  42990