Theorem hbim 2299

Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. 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 11	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
4	1, 3	hbim1 2297	1 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-ex 1786  df-nf 1790

This theorem is referenced by:  axi5r  2702  hbral  3146