Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  spimt Structured version   Visualization version   GIF version

Theorem spimt 2289
 Description: Closed theorem form of spim 2290. (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 ax6e 2286 . . . 4 𝑥 𝑥 = 𝑦
2 exim 1801 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑𝜓)))
31, 2mpi 20 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ∃𝑥(𝜑𝜓))
4 19.35 1845 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
53, 4sylib 208 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓))
6 19.9t 2109 . . 3 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
76biimpd 219 . 2 (Ⅎ𝑥𝜓 → (∃𝑥𝜓𝜓))
85, 7sylan9r 691 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1521  ∃wex 1744  Ⅎwnf 1748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator