Theorem dedlem0a

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)

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

Proof

Step	Hyp	Ref	Expression
1		iba	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜓 ∧ 𝜑 ) ) )
2		biimt	⊢ ( ( 𝜒 → 𝜑 ) → ( ( 𝜓 ∧ 𝜑 ) ↔ ( ( 𝜒 → 𝜑 ) → ( 𝜓 ∧ 𝜑 ) ) ) )
3	2	jarri	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜑 ) ↔ ( ( 𝜒 → 𝜑 ) → ( 𝜓 ∧ 𝜑 ) ) ) )
4	1 3	bitrd	⊢ ( 𝜑 → ( 𝜓 ↔ ( ( 𝜒 → 𝜑 ) → ( 𝜓 ∧ 𝜑 ) ) ) )