Theorem bj-exlimd 33979

Description: A slightly more general exlimd 2217. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2217. (Contributed by BJ, 25-Dec-2023.)

Hypotheses

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

Assertion

Ref	Expression
bj-exlimd	⊢ (𝜑 → (∃𝑥𝜒 → 𝜏))

Proof of Theorem bj-exlimd

Step	Hyp	Ref	Expression
1		bj-exlimd.th	. 2 ⊢ (𝜑 → (∃𝑥𝜃 → 𝜏))
2		bj-exlimd.ph	. . 3 ⊢ (𝜑 → ∀𝑥𝜓)
3		bj-exlimd.maj	. . 3 ⊢ (𝜓 → (𝜒 → 𝜃))
4	2, 3	sylg 1822	. 2 ⊢ (𝜑 → ∀𝑥(𝜒 → 𝜃))
5		bj-exlimg 33977	. 2 ⊢ ((∃𝑥𝜃 → 𝜏) → (∀𝑥(𝜒 → 𝜃) → (∃𝑥𝜒 → 𝜏)))
6	1, 4, 5	sylc 65	1 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜏))

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779

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

This theorem depends on definitions:  df-bi 209  df-ex 1780

This theorem is referenced by:  copsex2d  34455