Theorem ecased 643

Description: Deduction for elimination by cases.

Hypotheses

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

Assertion

Ref	Expression
ecased	⊢ (φ → ψ)

Proof of Theorem ecased

Step	Hyp	Ref	Expression
1		ecased.2	. . . 4 ⊢ (φ → ¬ χ)
2		ecased.3	. . . 4 ⊢ (φ → ¬ θ)
3	1, 2	jca 236	. . 3 ⊢ (φ → (¬ χ ∧ ¬ θ))
4		ioran 254	. . 3 ⊢ (¬ (χ ∨ θ) ↔ (¬ χ ∧ ¬ θ))
5	3, 4	sylibr 175	. 2 ⊢ (φ → ¬ (χ ∨ θ))
6		ecased.1	. . . 4 ⊢ (φ → (ψ ∨ χ ∨ θ))
7		3orass 584	. . . 4 ⊢ ((ψ ∨ χ ∨ θ) ↔ (ψ ∨ (χ ∨ θ)))
8	6, 7	sylib 173	. . 3 ⊢ (φ → (ψ ∨ (χ ∨ θ)))
9	8	ord 202	. 2 ⊢ (φ → (¬ ψ → (χ ∨ θ)))
10	5, 9	mt3d 101	1 ⊢ (φ → ψ)

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   ∨ w3o 580

This theorem is referenced by:  tz7.7 2224

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  df-3or 582

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