Theorem dedlema 920

Description: Lemma for weak deduction theorem. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Assertion

Ref	Expression
dedlema	⊢ (φ → (ψ ↔ ((ψ ∧ φ) ∨ (χ ∧ ¬ φ))))

Proof of Theorem dedlema

Step	Hyp	Ref	Expression
1		orc 374	. . 3 ⊢ ((ψ ∧ φ) → ((ψ ∧ φ) ∨ (χ ∧ ¬ φ)))
2	1	expcom 424	. 2 ⊢ (φ → (ψ → ((ψ ∧ φ) ∨ (χ ∧ ¬ φ))))
3		simpl 443	. . . 4 ⊢ ((ψ ∧ φ) → ψ)
4	3	a1i 10	. . 3 ⊢ (φ → ((ψ ∧ φ) → ψ))
5		pm2.24 101	. . . 4 ⊢ (φ → (¬ φ → ψ))
6	5	adantld 453	. . 3 ⊢ (φ → ((χ ∧ ¬ φ) → ψ))
7	4, 6	jaod 369	. 2 ⊢ (φ → (((ψ ∧ φ) ∨ (χ ∧ ¬ φ)) → ψ))
8	2, 7	impbid 183	1 ⊢ (φ → (ψ ↔ ((ψ ∧ φ) ∨ (χ ∧ ¬ φ))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by:  elimh  922  dedt  923  pm4.42  926