Theorem dedlem0a 1044

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 531	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))
2		biimt 364	. . 3 ⊢ ((𝜒 → 𝜑) → ((𝜓 ∧ 𝜑) ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑))))
3	2	jarri 107	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜑) ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑))))
4	1, 3	bitrd 282	1 ⊢ (𝜑 → (𝜓 ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399

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 210  df-an 400

This theorem is referenced by:  iftrue  4431