Theorem bj-alrimdh 37245

Description: Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2242 and 19.21h 2321. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) State the most general derivable instance. (Revised by BJ, 5-Apr-2026.)

Hypotheses

Ref	Expression
bj-alrimdh.nf1	⊢ (𝜑 → ∀𝑥𝜓)
bj-alrimdh.nf2	⊢ (𝜒 → ∀𝑥𝜃)
bj-alrimdh.maj	⊢ (𝜓 → (𝜃 → 𝜏))

Assertion

Ref	Expression
bj-alrimdh	⊢ (𝜑 → (𝜒 → ∀𝑥𝜏))

Proof of Theorem bj-alrimdh

Step	Hyp	Ref	Expression
1		bj-alrimdh.nf2	. 2 ⊢ (𝜒 → ∀𝑥𝜃)
2		bj-alrimdh.nf1	. . 3 ⊢ (𝜑 → ∀𝑥𝜓)
3		bj-alrimdh.maj	. . 3 ⊢ (𝜓 → (𝜃 → 𝜏))
4	2, 3	bj-alimdh 37244	. 2 ⊢ (𝜑 → (∀𝑥𝜃 → ∀𝑥𝜏))
5	1, 4	syl5 35	1 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1824  ax-4 1838

This theorem is used by: (None)