Theorem alimdh 1846

Description: Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1839. (Contributed by NM, 4-Jan-2002.)

Hypotheses

Ref	Expression
alimdh.1	⊢ (𝜑 → ∀𝑥𝜑)
alimdh.2	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

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

Proof of Theorem alimdh

Step	Hyp	Ref	Expression
1		alimdh.1	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		alimdh.2	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	al2imi 1844	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	1, 3	syl 18	1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1824  ax-4 1838

This theorem is used by:  alrimdh  1892  alimdv  1945  hbald  2202  alimd  2247  bj-cbvalimdlem  37279  bj-cbveximdlem  37280  bj-nfald  37807  dral1-o  39706  ax12indalem  39747  ax12inda2ALT  39748