Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elunif Structured version   Visualization version   GIF version

Theorem elunif 40837
Description: A version of eluni 4752 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
elunif.1 𝑥𝐴
elunif.2 𝑥𝐵
Assertion
Ref Expression
elunif (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
Distinct variable group:   𝐴,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem elunif
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eluni 4752 . 2 (𝐴 𝐵 ↔ ∃𝑦(𝐴𝑦𝑦𝐵))
2 elunif.1 . . . . 5 𝑥𝐴
3 nfcv 2949 . . . . 5 𝑥𝑦
42, 3nfel 2961 . . . 4 𝑥 𝐴𝑦
5 elunif.2 . . . . 5 𝑥𝐵
63, 5nfel 2961 . . . 4 𝑥 𝑦𝐵
74, 6nfan 1881 . . 3 𝑥(𝐴𝑦𝑦𝐵)
8 nfv 1892 . . 3 𝑦(𝐴𝑥𝑥𝐵)
9 eleq2w 2866 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
10 eleq1w 2865 . . . 4 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
119, 10anbi12d 630 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦𝐵) ↔ (𝐴𝑥𝑥𝐵)))
127, 8, 11cbvexv1 2321 . 2 (∃𝑦(𝐴𝑦𝑦𝐵) ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
131, 12bitri 276 1 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1761  wcel 2081  wnfc 2933   cuni 4749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-uni 4750
This theorem is referenced by:  eluni2f  40934  stoweidlem46  41899  stoweidlem57  41910
  Copyright terms: Public domain W3C validator