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

Theorem speiv 1785
Description: Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
Hypotheses
Ref Expression
speiv.1 (𝑥 = 𝑦 → (𝜑𝜓))
speiv.2 𝜓
Assertion
Ref Expression
speiv 𝑥𝜑
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem speiv
StepHypRef Expression
1 speiv.2 . 2 𝜓
2 speiv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimprd 156 . . 3 (𝑥 = 𝑦 → (𝜓𝜑))
43spimev 1784 . 2 (𝜓 → ∃𝑥𝜑)
51, 4ax-mp 7 1 𝑥𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator