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Theorem 19.23t 1640
Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
Assertion
Ref Expression
19.23t (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Proof of Theorem 19.23t
StepHypRef Expression
1 exim 1563 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
2 19.9t 1606 . . . 4 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
32biimpd 143 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
41, 3syl9r 73 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓)))
5 nfr 1483 . . . 4 (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
65imim2d 54 . . 3 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
7 19.38 1639 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
86, 7syl6 33 . 2 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑𝜓) → ∀𝑥(𝜑𝜓)))
94, 8impbid 128 1 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314  wnf 1421  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422
This theorem is referenced by:  19.23  1641  r19.23t  2516  ceqsalt  2686  vtoclgft  2710  sbciegft  2911
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