Theorem dedlem0b

Description: Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994)

	Ref	Expression
Assertion	dedlem0b	⊢ ( ¬ 𝜑 → ( 𝜓 ↔ ( ( 𝜓 → 𝜑 ) → ( 𝜒 ∧ 𝜑 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm2.21	⊢ ( ¬ 𝜑 → ( 𝜑 → ( 𝜒 ∧ 𝜑 ) ) )
2	1	imim2d	⊢ ( ¬ 𝜑 → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → ( 𝜒 ∧ 𝜑 ) ) ) )
3	2	com23	⊢ ( ¬ 𝜑 → ( 𝜓 → ( ( 𝜓 → 𝜑 ) → ( 𝜒 ∧ 𝜑 ) ) ) )
4		pm2.21	⊢ ( ¬ 𝜓 → ( 𝜓 → 𝜑 ) )
5		simpr	⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜑 )
6	4 5	imim12i	⊢ ( ( ( 𝜓 → 𝜑 ) → ( 𝜒 ∧ 𝜑 ) ) → ( ¬ 𝜓 → 𝜑 ) )
7	6	con1d	⊢ ( ( ( 𝜓 → 𝜑 ) → ( 𝜒 ∧ 𝜑 ) ) → ( ¬ 𝜑 → 𝜓 ) )
8	7	com12	⊢ ( ¬ 𝜑 → ( ( ( 𝜓 → 𝜑 ) → ( 𝜒 ∧ 𝜑 ) ) → 𝜓 ) )
9	3 8	impbid	⊢ ( ¬ 𝜑 → ( 𝜓 ↔ ( ( 𝜓 → 𝜑 ) → ( 𝜒 ∧ 𝜑 ) ) ) )