Theorem alimdh 1516

Description: Deduction from Theorem 19.20 of [Margaris] p. 90. (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 1507	. 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
4	1, 3	syl 14	1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1496  ax-gen 1498

This theorem is referenced by:  alrimdh  1528  hbald  1540  alimd  1570  hbimd  1622  dral1  1779  sbiedh  1836  ax16  1862  ax16i  1907  alimdv  1928