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Theorem 19.23t 1655
 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 1578 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
2 19.9t 1621 . . . 4 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
32biimpd 143 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
41, 3syl9r 73 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓)))
5 nfr 1498 . . . 4 (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
65imim2d 54 . . 3 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
7 19.38 1654 . . 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 1329  Ⅎwnf 1436  ∃wex 1468 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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437 This theorem is referenced by:  19.23  1656  r19.23t  2539  ceqsalt  2712  vtoclgft  2736  sbciegft  2939
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