Theorem hbim

Description: If x is not free in ph and ps , it is not free in ( ph -> ps ) . (Contributed by NM, 24-Jan-1993) (Proof shortened by Mel L. O'Cat, 3-Mar-2008) (Proof shortened by Wolf Lammen, 1-Jan-2018)

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

Proof

Step	Hyp	Ref	Expression
1		hbim.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		hbim.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3	2	a1i	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 𝜓 ) )
4	1 3	hbim1	⊢ ( ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) )