Theorem hbim1

Description: A closed form of hbim . (Contributed by NM, 2-Jun-1993)

	Ref	Expression
Hypotheses	hbim1.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	hbim1.2	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 𝜓 ) )
Assertion	hbim1	⊢ ( ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		hbim1.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		hbim1.2	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 𝜓 ) )
3	2	a2i	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 𝜓 ) )
4	1	19.21h	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) )
5	3 4	sylibr	⊢ ( ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) )