Theorem alimd 2205

Description: Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1813. See alimdh 1820, alimdv 1919 for variants requiring fewer axioms. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
alimd.1	⊢ Ⅎ𝑥𝜑
alimd.2	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
alimd	⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Proof of Theorem alimd

Step	Hyp	Ref	Expression
1		alimd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nf5ri 2188	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		alimd.2	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	2, 3	alimdh 1820	1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1537  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787

This theorem is referenced by:  alrimdd  2207  nfald  2322  mo3  2564  2mo  2650  axpowndlem3  10355  axextbdist  33776  pm11.71  42015