Theorem alrimdd 2214

Description: Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2207. (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	nf5rd 2196	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3		alrimdd.1	. . 3 ⊢ Ⅎ𝑥𝜑
4		alrimdd.3	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	3, 4	alimd 2212	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
6	2, 5	syld 47	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by:  alrimd  2215  wl-euequf  37575