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

Theorem spime 1720
 Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
Hypotheses
Ref Expression
spime.1 𝑥𝜑
spime.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spime (𝜑 → ∃𝑥𝜓)

Proof of Theorem spime
StepHypRef Expression
1 spime.1 . . . 4 𝑥𝜑
21a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
3 spime.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3spimed 1719 . 2 (⊤ → (𝜑 → ∃𝑥𝜓))
54mptru 1341 1 (𝜑 → ∃𝑥𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  ⊤wtru 1333  Ⅎwnf 1437  ∃wex 1469 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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-i9 1511  ax-ial 1515 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438 This theorem is referenced by:  spimev  1834
 Copyright terms: Public domain W3C validator