Theorem alrimdd 1602

Description: Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
alrimdd.1	⊢ Ⅎ𝑥𝜑
alrimdd.2	⊢ (𝜑 → Ⅎ𝑥𝜓)
alrimdd.3	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

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

Proof of Theorem alrimdd

Step	Hyp	Ref	Expression
1		alrimdd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
2	1	nfrd 1513	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3		alrimdd.1	. . 3 ⊢ Ⅎ𝑥𝜑
4		alrimdd.3	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	3, 4	alimd 1514	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
6	2, 5	syld 45	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1346  Ⅎwnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1440  ax-gen 1442  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  alrimd  1603