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Theorem bj-exlimd 34032
Description: A slightly more general exlimd 2219. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2219. (Contributed by BJ, 25-Dec-2023.)
Hypotheses
Ref Expression
bj-exlimd.ph (𝜑 → ∀𝑥𝜓)
bj-exlimd.th (𝜑 → (∃𝑥𝜃𝜏))
bj-exlimd.maj (𝜓 → (𝜒𝜃))
Assertion
Ref Expression
bj-exlimd (𝜑 → (∃𝑥𝜒𝜏))

Proof of Theorem bj-exlimd
StepHypRef Expression
1 bj-exlimd.th . 2 (𝜑 → (∃𝑥𝜃𝜏))
2 bj-exlimd.ph . . 3 (𝜑 → ∀𝑥𝜓)
3 bj-exlimd.maj . . 3 (𝜓 → (𝜒𝜃))
42, 3sylg 1824 . 2 (𝜑 → ∀𝑥(𝜒𝜃))
5 bj-exlimg 34030 . 2 ((∃𝑥𝜃𝜏) → (∀𝑥(𝜒𝜃) → (∃𝑥𝜒𝜏)))
61, 4, 5sylc 65 1 (𝜑 → (∃𝑥𝜒𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  copsex2d  34515
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