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Theorem exlimd2 1531
Description: Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1532 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)
Hypotheses
Ref Expression
exlimd2.1 (𝜑 → ∀𝑥𝜑)
exlimd2.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
exlimd2.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
exlimd2 (𝜑 → (∃𝑥𝜓𝜒))

Proof of Theorem exlimd2
StepHypRef Expression
1 exlimd2.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 exlimd2.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
31, 2alrimih 1403 . 2 (𝜑 → ∀𝑥(𝜒 → ∀𝑥𝜒))
4 exlimd2.3 . . 3 (𝜑 → (𝜓𝜒))
51, 4alrimih 1403 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
6 19.23ht 1431 . . 3 (∀𝑥(𝜒 → ∀𝑥𝜒) → (∀𝑥(𝜓𝜒) ↔ (∃𝑥𝜓𝜒)))
76biimpd 142 . 2 (∀𝑥(𝜒 → ∀𝑥𝜒) → (∀𝑥(𝜓𝜒) → (∃𝑥𝜓𝜒)))
83, 5, 7sylc 61 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-5 1381  ax-gen 1383  ax-ie2 1428
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  equsexd  1664  cbvexdh  1849
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