Theorem al2imi 1504

Description: Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
al2imi.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
al2imi	⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Proof of Theorem al2imi

Step	Hyp	Ref	Expression
1		al2imi.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alimi 1501	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒))
3		alim 1503	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒))
4	2, 3	syl 14	1 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1493  ax-gen 1495

This theorem is referenced by:  alanimi  1505  alimdh  1513  albi  1514  19.30dc  1673  19.33b2  1675  hbnt  1699  ax10o  1761  spimth  1781  sbi1v  1938  ralim  2589  ceqsalt  2826  intss  3944