Theorem hbim1 2305

Description: A closed form of hbim 2307. (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 2295	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
5	2, 4	sylibr 236	1 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785

This theorem is referenced by:  hbim  2307  axc14  2486