Theorem dedlem0a 1000

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 524	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))
2		ax-1 6	. . 3 ⊢ (𝜑 → (𝜒 → 𝜑))
3		biimt 350	. . 3 ⊢ ((𝜒 → 𝜑) → ((𝜓 ∧ 𝜑) ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑))))
4	2, 3	syl 17	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜑) ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑))))
5	1, 4	bitrd 268	1 ⊢ (𝜑 → (𝜓 ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384

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 197  df-an 386

This theorem is referenced by:  iftrue  4092