Theorem dedlem0a 970

Description: Alternate version of dedlema 971. (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 300	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))
2		ax-1 6	. . 3 ⊢ (𝜑 → (𝜒 → 𝜑))
3		biimt 241	. . 3 ⊢ ((𝜒 → 𝜑) → ((𝜓 ∧ 𝜑) ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑))))
4	2, 3	syl 14	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜑) ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑))))
5	1, 4	bitrd 188	1 ⊢ (𝜑 → (𝜓 ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)