Theorem hbimd

Description: Deduction form of bound-variable hypothesis builder hbim . (Contributed by NM, 14-May-1993) (Proof shortened by Wolf Lammen, 3-Jan-2018)

	Ref	Expression
Hypotheses	hbimd.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	hbimd.2	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 𝜓 ) )
	hbimd.3	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
Assertion	hbimd	⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → ∀ 𝑥 ( 𝜓 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hbimd.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		hbimd.2	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 𝜓 ) )
3		hbimd.3	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
4	1 2	nf5dh	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
5	1 3	nf5dh	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
6	4 5	nfimd	⊢ ( 𝜑 → Ⅎ 𝑥 ( 𝜓 → 𝜒 ) )
7	6	nf5rd	⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → ∀ 𝑥 ( 𝜓 → 𝜒 ) ) )