Theorem 4casesdan 1042

Description: Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.)

Hypotheses

Ref	Expression
4casesdan.1	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
4casesdan.2	⊢ ((𝜑 ∧ (𝜓 ∧ ¬ 𝜒)) → 𝜃)
4casesdan.3	⊢ ((𝜑 ∧ (¬ 𝜓 ∧ 𝜒)) → 𝜃)
4casesdan.4	⊢ ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃)

Assertion

Ref	Expression
4casesdan	⊢ (𝜑 → 𝜃)

Proof of Theorem 4casesdan

Step	Hyp	Ref	Expression
1		4casesdan.1	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
2	1	expcom 413	. 2 ⊢ ((𝜓 ∧ 𝜒) → (𝜑 → 𝜃))
3		4casesdan.2	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ ¬ 𝜒)) → 𝜃)
4	3	expcom 413	. 2 ⊢ ((𝜓 ∧ ¬ 𝜒) → (𝜑 → 𝜃))
5		4casesdan.3	. . 3 ⊢ ((𝜑 ∧ (¬ 𝜓 ∧ 𝜒)) → 𝜃)
6	5	expcom 413	. 2 ⊢ ((¬ 𝜓 ∧ 𝜒) → (𝜑 → 𝜃))
7		4casesdan.4	. . 3 ⊢ ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃)
8	7	expcom 413	. 2 ⊢ ((¬ 𝜓 ∧ ¬ 𝜒) → (𝜑 → 𝜃))
9	2, 4, 6, 8	4cases 1041	1 ⊢ (𝜑 → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395

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

This theorem is referenced by:  unxpdomlem3  9288  mndifsplit  22642  cdleme41snaw  40478  dihord  41266  dihjat  41425  2itscp  48702