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

Theorem spimt 1697
Description: Closed theorem form of spim 1699. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
Assertion
Ref Expression
spimt ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))

Proof of Theorem spimt
StepHypRef Expression
1 a9e 1657 . . . 4 𝑥 𝑥 = 𝑦
2 exim 1561 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑𝜓)))
31, 2mpi 15 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ∃𝑥(𝜑𝜓))
4 19.35-1 1586 . . 3 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
53, 4syl 14 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓))
6 19.9t 1604 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
76biimpd 143 . 2 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
85, 7sylan9r 405 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312   = wceq 1314  wnf 1419  wex 1451
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420
This theorem is referenced by:  spimd  12774
  Copyright terms: Public domain W3C validator