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

Theorem elunif 42008
 Description: A version of eluni 4799 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 4799 . 2 (𝐴 𝐵 ↔ ∃𝑦(𝐴𝑦𝑦𝐵))
2 elunif.1 . . . . 5 𝑥𝐴
3 nfcv 2920 . . . . 5 𝑥𝑦
42, 3nfel 2934 . . . 4 𝑥 𝐴𝑦
5 elunif.2 . . . . 5 𝑥𝐵
63, 5nfel 2934 . . . 4 𝑥 𝑦𝐵
74, 6nfan 1901 . . 3 𝑥(𝐴𝑦𝑦𝐵)
8 nfv 1916 . . 3 𝑦(𝐴𝑥𝑥𝐵)
9 eleq2w 2836 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
10 eleq1w 2835 . . . 4 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
119, 10anbi12d 634 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦𝐵) ↔ (𝐴𝑥𝑥𝐵)))
127, 8, 11cbvexv1 2352 . 2 (∃𝑦(𝐴𝑦𝑦𝐵) ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
131, 12bitri 278 1 (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1782   ∈ wcel 2112  Ⅎwnfc 2900  ∪ cuni 4796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-v 3412  df-uni 4797 This theorem is referenced by:  eluni2f  42102  stoweidlem46  43044  stoweidlem57  43055
 Copyright terms: Public domain W3C validator