Theorem alrimdh 1408

Description: Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Hypotheses

Ref	Expression
alrimdh.1	⊢ (𝜑 → ∀𝑥𝜑)
alrimdh.2	⊢ (𝜓 → ∀𝑥𝜓)
alrimdh.3	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
alrimdh	⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))

Proof of Theorem alrimdh

Step	Hyp	Ref	Expression
1		alrimdh.2	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2		alrimdh.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		alrimdh.3	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
4	2, 3	alimdh 1396	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
5	1, 4	syl5 32	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-5 1376  ax-gen 1378

This theorem is referenced by:  nfimd  1517  alrimdv  1797  moexexdc  2025