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

Theorem euabex 4221
Description: The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
euabex (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)

Proof of Theorem euabex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3660 . 2 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
2 vex 2740 . . . . 5 𝑦 ∈ V
32snex 4182 . . . 4 {𝑦} ∈ V
4 eleq1 2240 . . . 4 ({𝑥𝜑} = {𝑦} → ({𝑥𝜑} ∈ V ↔ {𝑦} ∈ V))
53, 4mpbiri 168 . . 3 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
65exlimiv 1598 . 2 (∃𝑦{𝑥𝜑} = {𝑦} → {𝑥𝜑} ∈ V)
71, 6sylbi 121 1 (∃!𝑥𝜑 → {𝑥𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wex 1492  ∃!weu 2026  wcel 2148  {cab 2163  Vcvv 2737  {csn 3591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator