Theorem hbim1 2297

Description: A closed form of hbim 2299. (Contributed by NM, 2-Jun-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 14	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
3		hbim1.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	19.21h 2287	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
5	2, 4	sylibr 233	1 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-ex 1784  df-nf 1788

This theorem is referenced by:  hbim  2299  axc14  2463