Theorem alimdh 1825

Description: Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1818. (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 1823	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	1, 3	syl 17	1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1803  ax-4 1817

This theorem is referenced by:  alrimdh  1871  alimdv  1924  hbald  2174  alimd  2212  bj-nfald  34992  dral1-o  36604  ax12indalem  36645  ax12inda2ALT  36646