ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.23t GIF version

Theorem 19.23t 1677
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 1599 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
2 19.9t 1642 . . . 4 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
32biimpd 144 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
41, 3syl9r 73 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓)))
5 nfr 1518 . . . 4 (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
65imim2d 54 . . 3 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
7 19.38 1676 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
86, 7syl6 33 . 2 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑𝜓) → ∀𝑥(𝜑𝜓)))
94, 8impbid 129 1 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351  wnf 1460  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  19.23  1678  r19.23t  2584  ceqsalt  2763  vtoclgft  2787  sbciegft  2993
  Copyright terms: Public domain W3C validator