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

Theorem spei 2398
Description: Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
spei.1 (𝑥 = 𝑦 → (𝜑𝜓))
spei.2 𝜓
Assertion
Ref Expression
spei 𝑥𝜑

Proof of Theorem spei
StepHypRef Expression
1 ax6e 2387 . 2 𝑥 𝑥 = 𝑦
2 spei.2 . . 3 𝜓
3 spei.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3mpbiri 248 . 2 (𝑥 = 𝑦𝜑)
51, 4eximii 1905 1 𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wex 1845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-12 2188  ax-13 2383
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1846
This theorem is referenced by:  elirrv  8658  bnj1014  31329
  Copyright terms: Public domain W3C validator