Theorem dedlem0a 918

Description: Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)

Assertion

Ref	Expression
dedlem0a	⊢ (φ → (ψ ↔ ((χ → φ) → (ψ ∧ φ))))

Proof of Theorem dedlem0a

Step	Hyp	Ref	Expression
1		iba 489	. 2 ⊢ (φ → (ψ ↔ (ψ ∧ φ)))
2		ax-1 6	. . 3 ⊢ (φ → (χ → φ))
3		biimt 325	. . 3 ⊢ ((χ → φ) → ((ψ ∧ φ) ↔ ((χ → φ) → (ψ ∧ φ))))
4	2, 3	syl 15	. 2 ⊢ (φ → ((ψ ∧ φ) ↔ ((χ → φ) → (ψ ∧ φ))))
5	1, 4	bitrd 244	1 ⊢ (φ → (ψ ↔ ((χ → φ) → (ψ ∧ φ))))

Syntax hints:   → wi 4   ↔ wb 176   ∧ 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-an 360

This theorem is referenced by:  iftrue  3669