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Theorem 19.24 2066
Description: Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
Assertion
Ref Expression
19.24 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Proof of Theorem 19.24
StepHypRef Expression
1 19.2 2058 . . 3 (∀𝑥𝜓 → ∃𝑥𝜓)
21imim2i 16 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
3 19.35 1954 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
42, 3sylibr 224 1 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-6 2054
This theorem depends on definitions:  df-bi 197  df-ex 1854
This theorem is referenced by: (None)
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