Theorem ecase3d 560

Description: Deduction for elimination by cases.

Hypotheses

Ref	Expression
ecase3d.1	⊢ (φ → (ψ → θ))
ecase3d.2	⊢ (φ → (χ → θ))
ecase3d.3	⊢ (φ → (¬ (ψ ∨ χ) → θ))

Assertion

Ref	Expression
ecase3d	⊢ (φ → θ)

Proof of Theorem ecase3d

Step	Hyp	Ref	Expression
1		ecase3d.1	. . 3 ⊢ (φ → (ψ → θ))
2	1	com12 13	. 2 ⊢ (ψ → (φ → θ))
3		ecase3d.2	. . 3 ⊢ (φ → (χ → θ))
4	3	com12 13	. 2 ⊢ (χ → (φ → θ))
5		ecase3d.3	. . 3 ⊢ (φ → (¬ (ψ ∨ χ) → θ))
6	5	com12 13	. 2 ⊢ (¬ (ψ ∨ χ) → (φ → θ))
7	2, 4, 6	ecase3 559	1 ⊢ (φ → θ)

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195

This theorem is referenced by:  distrlem4pr 3924  atcvat4 5775

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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