Theorem hbim 1817

Description: If x is not free in φ and ψ, it is not free in (φ → ψ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Hypotheses

Ref	Expression
hbim.1	⊢ (φ → ∀xφ)
hbim.2	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
hbim	⊢ ((φ → ψ) → ∀x(φ → ψ))

Proof of Theorem hbim

Step	Hyp	Ref	Expression
1		hbim.1	. 2 ⊢ (φ → ∀xφ)
2		hbim.2	. . 3 ⊢ (ψ → ∀xψ)
3	2	a1i 10	. 2 ⊢ (φ → (ψ → ∀xψ))
4	1, 3	hbim1 1810	1 ⊢ ((φ → ψ) → ∀x(φ → ψ))

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.23hOLD  1820  hbanOLD  1829  19.21hOLD  1840  cbv3hvOLD  1851  ax12olem5  1931  axi5r  2326  cleqh  2450  hbral  2663