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Theorem elunif 41150
Description: A version of eluni 4833 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 4833 . 2 (𝐴 𝐵 ↔ ∃𝑦(𝐴𝑦𝑦𝐵))
2 elunif.1 . . . . 5 𝑥𝐴
3 nfcv 2974 . . . . 5 𝑥𝑦
42, 3nfel 2989 . . . 4 𝑥 𝐴𝑦
5 elunif.2 . . . . 5 𝑥𝐵
63, 5nfel 2989 . . . 4 𝑥 𝑦𝐵
74, 6nfan 1891 . . 3 𝑥(𝐴𝑦𝑦𝐵)
8 nfv 1906 . . 3 𝑦(𝐴𝑥𝑥𝐵)
9 eleq2w 2893 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
10 eleq1w 2892 . . . 4 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
119, 10anbi12d 630 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦𝐵) ↔ (𝐴𝑥𝑥𝐵)))
127, 8, 11cbvexv1 2353 . 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 1771  wcel 2105  wnfc 2958   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-uni 4831
This theorem is referenced by:  eluni2f  41246  stoweidlem46  42208  stoweidlem57  42219
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