Theorem alrimdh 1790

Description: Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2075 and 19.21h 2121. (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 1745	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
5	1, 4	syl5 34	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))

Syntax hints:   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1722  ax-4 1737

This theorem is referenced by:  alrimdv  1857  ax12indn  34228  gen21nv  38845