Theorem dedlemb 921

Description: Lemma for weak deduction theorem. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Assertion

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

Proof of Theorem dedlemb

Step	Hyp	Ref	Expression
1		olc 373	. . 3 ⊢ ((χ ∧ ¬ φ) → ((ψ ∧ φ) ∨ (χ ∧ ¬ φ)))
2	1	expcom 424	. 2 ⊢ (¬ φ → (χ → ((ψ ∧ φ) ∨ (χ ∧ ¬ φ))))
3		pm2.21 100	. . . 4 ⊢ (¬ φ → (φ → χ))
4	3	adantld 453	. . 3 ⊢ (¬ φ → ((ψ ∧ φ) → χ))
5		simpl 443	. . . 4 ⊢ ((χ ∧ ¬ φ) → χ)
6	5	a1i 10	. . 3 ⊢ (¬ φ → ((χ ∧ ¬ φ) → χ))
7	4, 6	jaod 369	. 2 ⊢ (¬ φ → (((ψ ∧ φ) ∨ (χ ∧ ¬ φ)) → χ))
8	2, 7	impbid 183	1 ⊢ (¬ φ → (χ ↔ ((ψ ∧ φ) ∨ (χ ∧ ¬ φ))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by:  elimh  922  pm4.42  926  iffalse  3670