Theorem hbim1 1810

Description: A closed form of hbim 1817. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
hbim1.1	⊢ (φ → ∀xφ)
hbim1.2	⊢ (φ → (ψ → ∀xψ))

Assertion

Ref	Expression
hbim1	⊢ ((φ → ψ) → ∀x(φ → ψ))

Proof of Theorem hbim1

Step	Hyp	Ref	Expression
1		hbim1.2	. . 3 ⊢ (φ → (ψ → ∀xψ))
2	1	a2i 12	. 2 ⊢ ((φ → ψ) → (φ → ∀xψ))
3		hbim1.1	. . 3 ⊢ (φ → ∀xφ)
4	3	19.21h 1797	. 2 ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))
5	2, 4	sylibr 203	1 ⊢ ((φ → ψ) → ∀x(φ → ψ))

This proof depends on syntax axioms:   → wi 4  ∀wal 1540

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

This theorem is used by:  nfim1  1811  hbim  1817  ax12olem6  1932  ax15  2021