ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abn0m GIF version

Theorem abn0m 3440
Description: Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)
Assertion
Ref Expression
abn0m (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abn0m
StepHypRef Expression
1 nfv 1521 . . 3 𝑦 𝑥 ∈ {𝑥𝜑}
2 nfsab1 2160 . . 3 𝑥 𝑦 ∈ {𝑥𝜑}
3 eleq1w 2231 . . 3 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜑}))
41, 2, 3cbvex 1749 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 ∈ {𝑥𝜑})
5 abid 2158 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
65exbii 1598 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
74, 6bitr3i 185 1 (∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wex 1485  wcel 2141  {cab 2156
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-clel 2166
This theorem is referenced by:  mapprc  6630
  Copyright terms: Public domain W3C validator