This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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