Theorem bj-alrimd 33972

Description: A slightly more general alrimd 2214. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2214. (Contributed by BJ, 25-Dec-2023.)

Hypotheses

Ref	Expression
bj-alrimd.ph	⊢ (𝜑 → ∀𝑥𝜓)
bj-alrimd.th	⊢ (𝜑 → (𝜒 → ∀𝑥𝜃))
bj-alrimd.maj	⊢ (𝜓 → (𝜃 → 𝜏))

Assertion

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

Proof of Theorem bj-alrimd

Step	Hyp	Ref	Expression
1		bj-alrimd.th	. 2 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜃))
2		bj-alrimd.ph	. . 3 ⊢ (𝜑 → ∀𝑥𝜓)
3		bj-alrimd.maj	. . 3 ⊢ (𝜓 → (𝜃 → 𝜏))
4	2, 3	sylg 1822	. 2 ⊢ (𝜑 → ∀𝑥(𝜃 → 𝜏))
5		bj-alrimg 33971	. 2 ⊢ ((𝜒 → ∀𝑥𝜃) → (∀𝑥(𝜃 → 𝜏) → (𝜒 → ∀𝑥𝜏)))
6	1, 4, 5	sylc 65	1 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏))

Syntax hints:   → wi 4  ∀wal 1534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1795  ax-4 1809

This theorem is referenced by: (None)